8/28/10

Well this week was basically a review of what we learned at the end of the year last year in pre-calc. Last week we went over limits and the different ways you can find them. For instance, the first thing you can do if you are asked to find the limit of a function is plug in the number they give to you as x. That works only when you get a number after you solve it. If you get zero as the denominator, then you have to try a new method. Another way to find a limit is to factor. If you’re asked to find the limit of a function that is a fraction, like for instance (x+2)/(x^2+3x+2), you can factor and cancel and then plug in the number they give you as x. Another way to find a limit (also if you get zero as the denominator first) you can use the table in your calculator and plug in the correct values. This week we learned two more special ways to find limits.—rationalizing and the Squeeze Theorem, and how to discuss the continuity of functions. I thought rationalizing was really easy, so here’s an example problem where you’d have to use that to find the limit:

Ex. 1) Find the limit (if it exists)… x>3lim ((sqrt of (x+1) – 2) / (x-3)*sqrt=square root/x>3=x goes to 3

*All “rationalizing” really means is to simplify the problem more so that it’s easier to solve. You can use rationalizing when you have a fraction, a square root, and something being added or subtracted.
*So for this problem you can use rationalizing since you have a square root and something being subtracted in the numerator. So the first thing you do is multiply the numerator and denominator of the problem by the numerator’s reciprocal.>> (sqrt of x+1) + 2
*Then multiplying the top out, you get x+1-4
*And for the denominator you get (x-3)((sqrt of x+1) +2)
*Simplifying the top you get (x-3)…So you can cancel out the (x-3)’s on the top and bottom of the fraction. Now you should be left with >>> (1)/((sqrt of x+1)+2)
*Now you can plug in 3 into that equation to get the limit.
So you get, (1)/(2+2) which is 1/4

**I basically understood everything this week. The only thing I’m a little iffy with is the Intermediate Value Theorem. I looked at the example in my notes, but I’m still not sure how I would do a problem like that on my own. The part where you would have to look at the graph confuses me.