Explain how to find each type of discontinuity and provide an example.
a. Jump 1. You must completely factor the top and bottom of the fraction 2. Cancel all items that are able to be cancelled. 3. Set what is left in the denominator equal to 0 4. Solve for x If X=# can be plugged back into the equation and cause the fraction to have a zero as the denominator X=# is a jump
EX: f(x)= 1/x 1. Completely factored 2. No Items need to be cancelled 3. X=0 4. X=0 If zero is plugged back in for x the denominator would be 0 therefore at F(x)= 1/x there is a jump.
b. Removable ((Remember the problem must be a fraction or a piecewise)) if the problem is a fraction 1. You must completely factor the top and the bottom 2. Cancel all items that are able to be cancelled. 3. Set anything that was cancelled equal to 0 4. Solve for x X=# is the removable. Ex: F(x)= x^2-1/x-1 1. (x+1)(x-1)/(x-1) 2. (x+1)(x-1)/(x-1) 3. Because (x-1) cancelled set x-1 equal to 0 4. X-1=0 therefore x= 1 At x=1 there is a removable
If the problem is a piecewise 1. You complete the piecewise by plugging in for the x’s 2. Check that each component of the piecewise has the same y value X=# is the removable
Ex: f(x)= { x+1 x > 0 { x^2+ 1 x > 0
1. 0+1 = 1 0^2 + 1 = 1
2. Both y values match Therefore x= 1 is a removable
c. Vertical Asymptote The steps to finding a vertical asymptote are similar to the steps of finding a removable when the problem is a fraction. 1. You must factor the top and bottom completely. 2. Cancel all items that are able to be cancelled. 3. Set what is left in the denominator equal to 0 4. Solve for x X= # are vertical asymptotes
1.) Jump: The simplest way to recognize a jump is from the graph. If you see that there is a break in the graph, as in you would have to pick up your pencil to finish drawing it, then that's a jump. So that means that the graph is not continuous. Jumps are sometimes in functions called piecewises, and are ALWAYS (correct me if I'm wrong) present when it comes to an equation with the "greatest integer function" included. When it comes to piecewises, the first thing that has to be present is a greater than/equal to or less than/equal to sign (with the bar) If there is no bar, then there's usually no chance of there being a jump, but in some cases there may be. Then, once you plug the x value given to you into the equations you have, you should NOT get the same number. In order for there to be a jump, the numbers you get have to be different. Here's a few examples: A.) Piecewise function: (x+2)/2 x3 **x abs value of x+3/x+3
B.) lim x>4 (5[x]-7) *let's pretend that [ ] actually looks like the greatest integer function *Okay so to find the limit of this, as x goes to 4 all you have to do is plug in 4 and you get 13. But to know that there is a jump there, you can type it into your calculator and you would see that there are several jumps....or you could just remember that that's always what the greatest integer function's graph looks like
2.) Removable: To review, removables are "holes" in a graph, they are not continuous, however they ALWAYS have a limit. Removables can be found in fractions or in piecewises. For a fraction, you would have to factor the top and bottom, cancel whatever possible, and set what you canceled equal to zero. Now obviously if you didn't cancel anything, are you going to have a removable? Not a chance. As for piecewises, there CANNOT be a greater than/equal or less than/equal to sign (meaning there cannot be a BAR) However in some cases, even if there are no bars, there may be a jump instead. Here's an example: A.) (x+2)/(x^2-3x-10) *Okay the first thing you want to do is ask yourself if anything's factorable....Yes, you can factor the denominator to (x-5)(x+2) *Now you can cancel (x+2) from the top and bottom of the fraction. Then you set x+2 equal to zero to get your removable. *Sooooo there would be a removable at x=-2
3.) Vertical Asymptotes: Again vertical asymptotes are imaginary lines that exist on some graphs, and they SOMETIMES have a limit. Vertical asymptotes can be found in fractions and in these trig functions: tan(x), sec(x), cot(x), and csc(x)...because they all have undefined values in the trig chart. To see if there's a vertical asymptote in a fraction you first see if it is factorable. If so, you cancel whatever possible (which would be the removable) and whatever is left in the denominator would be the vertical asymptote once you set it equal to zero. Now if you can't factor or cancel anything in a fraction that means there's no vertical asymptote right? WRONGGGGG. If there is still something in the denominator (with an x), then you set it equal to zero and that becomes the vertical asymptote. Here's an example:
A.) Discuss the discontinuity of 1/(x^2-1) *First thing you can do is factor the bottom out to (x+1)(x-1)..*difference of squares *Nothing can be factored further and there's nothing to cancel, so you set x+1 and x-1 both equal to zero to get your vertical asymptotes. *So your vertical asymptotes are at x=1 and x=-1
Okay, now to explain how to find the following discontinuities:
a) Jump: A jump is usually in the form of a piecewise. In a piecewise, there are at least two equations. If you plug in the given x-value into the two equations and get two different answers, then there’s a jump. Another example of a jump is the greatest integer function (which kind of looks like double brackets around an x). No matter what, when you have a greatest integer function, there will always be a jump (well, actually, several jumps). Anyway, the easiest way to find a jump is to graph your function in your calculator. You can tell just by looking at the graph if there is a jump or not, easy as that.
Ex. *please picture the following two equations as a piecewise: lim as x approaches 1 : 2x, x < (or equal to) 1 & x^2, x > 1 Plug in 1 for x, and you get 2x = 2 & x^2 = 1 Clearly, the two are different, therefore, there is a jump at x=1
b) Removable: A removable is very easy to find. When you have an equation as a fraction and you plug in the x-value only to get 0 on the bottom, you have to factor. When you expand the equation as much as possible, you cancel. Now, the things you cancel out of the bottom you set equal to 0 and that’s your removable.
Ex. lim as x approaches 2 : x^2 – 4 / x – 2 First, factor. (x-2)(x+2) / (x-2) Now cancel, and set what you canceled equal to 0. The (x-2) cancels. x=2 is a removable
c) Vertical Asymptote: A vertical asymptote is just as easy to find as a removable. When you have an equation as a fraction and you plug in the x-value only to get 0 on the bottom, you have to factor. When you expand the equation as much as possible, you cancel. Once you’ve factored and cancelled, if there’s anything left in the denominator, set it equal to 0 and that’s your vertical asymptote. You can also find the vertical asymptote by plugging your equation into your calculator and looking at the graph.
Ex. lim as x approaches 3 : (x-2) / (x-3) First, factor. This one is already factored. Then, cancel if possible. Nothing cancels here. Then, set what’s left in the bottom equal to 0. x=3 is a vertical asymptote
Vertical Asymptotes: To have a vertical asymptotes, the graph must be undefined.So, that means it must be a fraction where the bottom equals zero or a trig function that equals is undefined somewhere. To find where the vertical asymptote is on the graph, first you must factor the top and bottom. After expanding, cancel all that can be canceled. Set whatever is left in the denominator equal to zero(if anything). Ex. x^2+4/x+2 Nothing can factor nor cancel. So, you set the bottom equal to zero and get -2. Therefore there is a vertical asymptote at x=-2.
Removables: If you have a fraction, it is possible there is a removable. First thing you do to check is factor the top and the bottom. Once factored, check to see if there is anything you can cancel. If so, set what you canceled equal to zero and there is a removable at that point or points. If there is nothing to cancel, or nothing to factor, there is no removable. Ex. x^2-4/x+2 Factor it and you find that the x+2 cancels. Set x+2 equal to zero and you find out there is a removable at x=-2
You can also find a removable from a piecewise. First thing you must check is if there is a bar, which means equal to, so no greater than or equal to or no less than or equal to.
Jumps: Jumps show up at piecewises. If there is a bar on one of your signs, then there is no removable, but it is possible there is a jump. To find the jump, if present, plug in your x values. If you get different y values as answers, then you win. Ex. f(x)={x+9 x=4 x/2 x=0 After plugging in your x's you get 13 and 0 as your y values showing that there is a jump in this equation.
a jump is a gap in a graph, usually in a piecewise. to check for jumps, plug in for x in the equation; if the answers dont match, you have a jump. also, if there is a fraction, factor as far as possible, cancel out what you can, set the bottom =0, and solve for x; if the result causes the bottom to still =0, theres a jump
example: lim x->2 x^3 - 4
plugging in 2, the values dont =, so there is a jump.
B) Removables
a removable is a hole in a graph that always has a limit; to find a removable: factor, cancel, set bottom =0, x = # is the removable
lim x-> 2 x^3
it can be factored (x+3)(x-3)/x-3 x-3 cancels out x=3 is removable
C) vertical asymptote
the lower x value causes it to =0 with direct substitution; factor, cancel out, set the bottom =0, x is the vertical asymptote
lim x->5 x+6/x-5 its already factored, cannot be cancelled, set bottom =0
We have to explain how to find the three different types of discontinuities. The three types of discontinuities are a jump, removable, and a vertical asymptote. To start, I will explain how to know if something is continuous or not. For something to be continuous the left and the right side must be approaching the same number, and if a limit exists, it must be equal to a function value at that point. However, jumps, removables, and vertical asymptotes are discontinuities meaning that both sides of the graph are not allowed to approach the same number.
1. Jump The easiest way to see if a function has a discontinuity is to graph it. When graphing it, if there is a jump in the graph, or a point when the graph will break from one number to another it would be classified as a jump. -How does one figure it out without looking at the graph? If the function is x=# and it takes on a different value, it is usually a jump. *A jump could also be put in a piecewise function.
2. Removable There are several ways to find out if a removable has a discontinuity. One way is to try and solve a function. You would first: factor and cancel. Then: take what you cancelled and set it equal to zero. Whatever answer you get (x=#) that would be the removable. In a graph, a removable discontinuity is a point at which a graph is not connected. There will be an open circle at the point where there is a removable.
3. Vertical Asymptote **Vertical asymptotes occur when it is undefined, infinity, or negative infinity. You can do two things to see if a vertical asymptote has a discontinuity. When graphing the function, the number that does not get touched by the vertical asymptotes will be the discontinuity. When solving without your calculator, follow the same steps as a removable. Except set what is left in the bottom equal to zero. x=# will be the discontinuity.
crap, I almost forgot about this, good thing for backtracing thoughts ahh, the good old discontinuities
1) jump: to find a jump, you have to........use a piecewise......and stuff.............and if the answers to the equations in the piecewise are different, then it's a jump...........also, if it's a fraction.......factor the top and bottom out completely......cross out what you can.......set the bottom equal to 0.......solve for x......plug x = # into the equation to see if the denominator ≠ 0.......if it does.....then you, my friend, have just found yourself a jump ex: lim x-4 x->7 plug in 7 for x and it doesn't equal 0, so it's a jump
2) removable: a removable is easy to find....all you have to do, when it's a fraction, is factor the top and bottom, and cross out whatever you can....whatever you cross out, you set it equal to 0 and solve it for x....whatever x = is a removable ex: lim x^2 -4/x^-x-6 x->3 you factor it out to be ((x+2)(x-2))/((x+3)(x-2)) cross out the x-2 from the bottom and top x-2=0 x=2 is a removable
3) vertical asymptote: vertical asymptotes are easy to find as well........follow the same steps as finding removables, except you set whatever you DIDN'T cross out in the denominator = 0, solve for x, and whatever x = is a vertical asymptote ex: lim x^2-4/x^-x-6 x->3 ((x+2)(x-2))/((x+3)(x-2)) cross out x-2 set x+3=0 x=-3 is a vertical asymptote
Explain how to find each type of discontinuity and provide an example.
Jump
First factor the top and bottom of the fraction, then cancel everything that can be canceled, then Set what is not canceled in the bottom of the fraction equal to 0, and finally solve for x. What x equals is where your jump is.
Ex: f(x)= x/x^2+1 it is factored already cancel a x whats left is x+1 jump is at x=-1
Removables
In a fraction you follow your same rules for a jump. First make sure that everything is factored. Then you must cancel anything that can be canceled. Instead of setting what is left equal to zero, you set what you canceled equal to zero.
Ex. f(x)=x/x^2+1 it is already factored you can cancel a x set the x equal to zero there is a removable at x=0
In a piecewise
If there is a piecewise you must plug in x to complete the piecewise. Then if you have the same y values then you have a removable.
Ex: f(x)= { x+1 x > 1 { x^2+ 1 x > 1
1. 1+1 = 2 1^2 + 1 = 2
the y values are the same remoavable at x= 1
Vertical asymptote
You follow the same stepts that you used when looking for a jump. Make sure that the fraction is factored. then cancel what you can. and whats left in the bottom, set equal to zero. this will be your vertical asymptote.
Ex: f(x)= x/ 3x-1) its already factored you can cancel an x your left with 3-1 set equal to zero 3-1=0 =2 vertical asymptote at 2
Explain how to find each type of discontinuity and provide an example.
ReplyDeletea. Jump
1. You must completely factor the top and bottom of the fraction
2. Cancel all items that are able to be cancelled.
3. Set what is left in the denominator equal to 0
4. Solve for x
If X=# can be plugged back into the equation and cause the fraction to have a zero as the denominator X=# is a jump
EX: f(x)= 1/x
1. Completely factored
2. No Items need to be cancelled
3. X=0
4. X=0
If zero is plugged back in for x the denominator would be 0 therefore at
F(x)= 1/x there is a jump.
b. Removable
((Remember the problem must be a fraction or a piecewise))
if the problem is a fraction
1. You must completely factor the top and the bottom
2. Cancel all items that are able to be cancelled.
3. Set anything that was cancelled equal to 0
4. Solve for x
X=# is the removable.
Ex: F(x)= x^2-1/x-1
1. (x+1)(x-1)/(x-1)
2. (x+1)(x-1)/(x-1)
3. Because (x-1) cancelled set x-1 equal to 0
4. X-1=0 therefore x= 1
At x=1 there is a removable
If the problem is a piecewise
1. You complete the piecewise by plugging in for the x’s
2. Check that each component of the piecewise has the same y value
X=# is the removable
Ex: f(x)= { x+1 x > 0
{ x^2+ 1 x > 0
1. 0+1 = 1
0^2 + 1 = 1
2. Both y values match
Therefore x= 1 is a removable
c. Vertical Asymptote
The steps to finding a vertical asymptote are similar to the steps of finding a removable when the problem is a fraction.
1. You must factor the top and bottom completely.
2. Cancel all items that are able to be cancelled.
3. Set what is left in the denominator equal to 0
4. Solve for x
X= # are vertical asymptotes
Ex: f(x)= 1/ 2(x+1)
1. 1/ 2x + 2
2. Nothing can be cancelled
3. 2x + 2 = 0
4. 2x + 2 = 0 therefore x= -1
X= -1 is the vertical asymptotes
PROMPT #2
ReplyDelete1.) Jump:
The simplest way to recognize a jump is from the graph. If you see that there is a break in the graph, as in you would have to pick up your pencil to finish drawing it, then that's a jump. So that means that the graph is not continuous. Jumps are sometimes in functions called piecewises, and are ALWAYS (correct me if I'm wrong) present when it comes to an equation with the "greatest integer function" included. When it comes to piecewises, the first thing that has to be present is a greater than/equal to or less than/equal to sign (with the bar) If there is no bar, then there's usually no chance of there being a jump, but in some cases there may be. Then, once you plug the x value given to you into the equations you have, you should NOT get the same number. In order for there to be a jump, the numbers you get have to be different. Here's a few examples:
A.) Piecewise function:
(x+2)/2 x3
**x abs value of x+3/x+3
B.) lim x>4 (5[x]-7)
*let's pretend that [ ] actually looks like the greatest integer function
*Okay so to find the limit of this, as x goes to 4 all you have to do is plug in 4 and you get 13. But to know that there is a jump there, you can type it into your calculator and you would see that there are several jumps....or you could just remember that that's always what the greatest integer function's graph looks like
2.) Removable:
To review, removables are "holes" in a graph, they are not continuous, however they ALWAYS have a limit. Removables can be found in fractions or in piecewises. For a fraction, you would have to factor the top and bottom, cancel whatever possible, and set what you canceled equal to zero. Now obviously if you didn't cancel anything, are you going to have a removable? Not a chance. As for piecewises, there CANNOT be a greater than/equal or less than/equal to sign (meaning there cannot be a BAR) However in some cases, even if there are no bars, there may be a jump instead. Here's an example:
A.) (x+2)/(x^2-3x-10)
*Okay the first thing you want to do is ask yourself if anything's factorable....Yes, you can factor the denominator to (x-5)(x+2)
*Now you can cancel (x+2) from the top and bottom of the fraction. Then you set x+2 equal to zero to get your removable.
*Sooooo there would be a removable at x=-2
3.) Vertical Asymptotes:
Again vertical asymptotes are imaginary lines that exist on some graphs, and they SOMETIMES have a limit. Vertical asymptotes can be found in fractions and in these trig functions: tan(x), sec(x), cot(x), and csc(x)...because they all have undefined values in the trig chart. To see if there's a vertical asymptote in a fraction you first see if it is factorable. If so, you cancel whatever possible (which would be the removable) and whatever is left in the denominator would be the vertical asymptote once you set it equal to zero. Now if you can't factor or cancel anything in a fraction that means there's no vertical asymptote right? WRONGGGGG. If there is still something in the denominator (with an x), then you set it equal to zero and that becomes the vertical asymptote. Here's an example:
A.) Discuss the discontinuity of 1/(x^2-1)
*First thing you can do is factor the bottom out to (x+1)(x-1)..*difference of squares
*Nothing can be factored further and there's nothing to cancel, so you set x+1 and x-1 both equal to zero to get your vertical asymptotes.
*So your vertical asymptotes are at x=1 and x=-1
Okay, now to explain how to find the following discontinuities:
ReplyDeletea) Jump: A jump is usually in the form of a piecewise. In a piecewise, there are at least two equations. If you plug in the given x-value into the two equations and get two different answers, then there’s a jump. Another example of a jump is the greatest integer function (which kind of looks like double brackets around an x). No matter what, when you have a greatest integer function, there will always be a jump (well, actually, several jumps). Anyway, the easiest way to find a jump is to graph your function in your calculator. You can tell just by looking at the graph if there is a jump or not, easy as that.
Ex. *please picture the following two equations as a piecewise: lim as x approaches 1 :
2x, x < (or equal to) 1 & x^2, x > 1
Plug in 1 for x, and you get 2x = 2 & x^2 = 1
Clearly, the two are different, therefore, there is a jump at x=1
b) Removable: A removable is very easy to find. When you have an equation as a fraction and you plug in the x-value only to get 0 on the bottom, you have to factor. When you expand the equation as much as possible, you cancel. Now, the things you cancel out of the bottom you set equal to 0 and that’s your removable.
Ex. lim as x approaches 2 : x^2 – 4 / x – 2
First, factor. (x-2)(x+2) / (x-2)
Now cancel, and set what you canceled equal to 0. The (x-2) cancels.
x=2 is a removable
c) Vertical Asymptote: A vertical asymptote is just as easy to find as a removable. When you have an equation as a fraction and you plug in the x-value only to get 0 on the bottom, you have to factor. When you expand the equation as much as possible, you cancel. Once you’ve factored and cancelled, if there’s anything left in the denominator, set it equal to 0 and that’s your vertical asymptote. You can also find the vertical asymptote by plugging your equation into your calculator and looking at the graph.
Ex. lim as x approaches 3 : (x-2) / (x-3)
First, factor. This one is already factored.
Then, cancel if possible. Nothing cancels here.
Then, set what’s left in the bottom equal to 0.
x=3 is a vertical asymptote
This comment has been removed by the author.
ReplyDeleteThis comment has been removed by the author.
ReplyDeleteVertical Asymptotes:
ReplyDeleteTo have a vertical asymptotes, the graph must be undefined.So, that means it must be a fraction where the bottom equals zero or a trig function that equals is undefined somewhere. To find where the vertical asymptote is on the graph, first you must factor the top and bottom. After expanding, cancel all that can be canceled. Set whatever is left in the denominator equal to zero(if anything).
Ex.
x^2+4/x+2
Nothing can factor nor cancel. So, you set the bottom equal to zero and get -2. Therefore there is a vertical asymptote at x=-2.
Removables:
If you have a fraction, it is possible there is a removable. First thing you do to check is factor the top and the bottom. Once factored, check to see if there is anything you can cancel. If so, set what you canceled equal to zero and there is a removable at that point or points. If there is nothing to cancel, or nothing to factor, there is no removable.
Ex.
x^2-4/x+2
Factor it and you find that the x+2 cancels.
Set x+2 equal to zero and you find out there is a removable at x=-2
You can also find a removable from a piecewise.
First thing you must check is if there is a bar, which means equal to, so no greater than or equal to or no less than or equal to.
Jumps:
Jumps show up at piecewises. If there is a bar on one of your signs, then there is no removable, but it is possible there is a jump. To find the jump, if present, plug in your x values. If you get different y values as answers, then you win.
Ex. f(x)={x+9 x=4
x/2 x=0
After plugging in your x's you get 13 and 0 as your y values showing that there is a jump in this equation.
A) Jumps
ReplyDeletea jump is a gap in a graph, usually in a piecewise. to check for jumps, plug in for x in the equation; if the answers dont match, you have a jump. also, if there is a fraction, factor as far as possible, cancel out what you can, set the bottom =0, and solve for x; if the result causes the bottom to still =0, theres a jump
example: lim x->2
x^3 - 4
plugging in 2, the values dont =, so there is a jump.
B) Removables
a removable is a hole in a graph that always has a limit; to find a removable: factor, cancel, set bottom =0, x = # is the removable
lim x-> 2
x^3
it can be factored (x+3)(x-3)/x-3
x-3 cancels out
x=3 is removable
C) vertical asymptote
the lower x value causes it to =0 with direct substitution; factor, cancel out, set the bottom =0, x is the vertical asymptote
lim x->5 x+6/x-5
its already factored, cannot be cancelled, set bottom =0
x=5 is a vertical asymptote
We have to explain how to find the three different types of discontinuities. The three types of discontinuities are a jump, removable, and a vertical asymptote.
ReplyDeleteTo start, I will explain how to know if something is continuous or not. For something to be continuous the left and the right side must be approaching the same number, and if a limit exists, it must be equal to a function value at that point. However, jumps, removables, and vertical asymptotes are discontinuities meaning that both sides of the graph are not allowed to approach the same number.
1. Jump
The easiest way to see if a function has a discontinuity is to graph it. When graphing it, if there is a jump in the graph, or a point when the graph will break from one number to another it would be classified as a jump.
-How does one figure it out without looking at the graph? If the function is x=# and it takes on a different value, it is usually a jump.
*A jump could also be put in a piecewise function.
2. Removable
There are several ways to find out if a removable has a discontinuity. One way is to try and solve a function.
You would first: factor and cancel.
Then: take what you cancelled and set it equal to zero. Whatever answer you get (x=#) that would be the removable.
In a graph, a removable discontinuity is a point at which a graph is not connected. There will be an open circle at the point where there is a removable.
3. Vertical Asymptote
**Vertical asymptotes occur when it is undefined, infinity, or negative infinity. You can do two things to see if a vertical asymptote has a discontinuity.
When graphing the function, the number that does not get touched by the vertical asymptotes will be the discontinuity. When solving without your calculator, follow the same steps as a removable. Except set what is left in the bottom equal to zero. x=# will be the discontinuity.
crap, I almost forgot about this, good thing for backtracing thoughts
ReplyDeleteahh, the good old discontinuities
1) jump: to find a jump, you have to........use a piecewise......and stuff.............and if the answers to the equations in the piecewise are different, then it's a jump...........also, if it's a fraction.......factor the top and bottom out completely......cross out what you can.......set the bottom equal to 0.......solve for x......plug x = # into the equation to see if the denominator ≠ 0.......if it does.....then you, my friend, have just found yourself a jump
ex: lim x-4
x->7
plug in 7 for x and it doesn't equal 0, so it's a jump
2) removable: a removable is easy to find....all you have to do, when it's a fraction, is factor the top and bottom, and cross out whatever you can....whatever you cross out, you set it equal to 0 and solve it for x....whatever x = is a removable
ex: lim x^2 -4/x^-x-6
x->3
you factor it out to be ((x+2)(x-2))/((x+3)(x-2))
cross out the x-2 from the bottom and top
x-2=0
x=2 is a removable
3) vertical asymptote: vertical asymptotes are easy to find as well........follow the same steps as finding removables, except you set whatever you DIDN'T cross out in the denominator = 0, solve for x, and whatever x = is a vertical asymptote
ex: lim x^2-4/x^-x-6
x->3
((x+2)(x-2))/((x+3)(x-2))
cross out x-2
set x+3=0
x=-3 is a vertical asymptote
Explain how to find each type of discontinuity and provide an example.
ReplyDeleteJump
First factor the top and bottom of the fraction, then cancel everything that can be canceled, then Set what is not canceled in the bottom of the fraction equal to 0, and finally solve for x. What x equals is where your jump is.
Ex: f(x)= x/x^2+1
it is factored already
cancel a x
whats left is x+1
jump is at x=-1
Removables
In a fraction you follow your same rules for a jump. First make sure that everything is factored. Then you must cancel anything that can be canceled. Instead of setting what is left equal to zero, you set what you canceled equal to zero.
Ex. f(x)=x/x^2+1
it is already factored
you can cancel a x
set the x equal to zero
there is a removable at x=0
In a piecewise
If there is a piecewise you must plug in x to complete the piecewise. Then if you have the same y values then you have a removable.
Ex: f(x)= { x+1 x > 1
{ x^2+ 1 x > 1
1. 1+1 = 2
1^2 + 1 = 2
the y values are the same
remoavable at x= 1
Vertical asymptote
You follow the same stepts that you used when looking for a jump. Make sure that the fraction is factored. then cancel what you can. and whats left in the bottom, set equal to zero. this will be your vertical asymptote.
Ex: f(x)= x/ 3x-1)
its already factored
you can cancel an x
your left with 3-1
set equal to zero
3-1=0
=2
vertical asymptote at 2