1. 3-2 Limits Limit as x approaches a number

2. Just what is a limit? A limit is what the y (that is, f(x)) value of a function appears to approach as x approaches a particular number. We may not actually get there. We may jump over it. BUT a limit is not what you actually get to, but appear to get to. Lets see what it means with graphing.graphing

3. What is the way to solve it? The easiest way to find the limit value is to plug the number in. Find the following

4. Uhoh… What about Lets go graphing and lookgraphing OK – so how will we actually get the value 4?

5. What if…. What do you think the answer is if you plug in the number and get ? What do you think the answer is if you plug in the number and get ? So the lesson here is that you ONLY have to factor top and bottom if you get.

6. How about this one? What do we do when we can’t plug the number in? Back to graphinggraphing

7. The Harmonic Sequence The process used to find limits as x  is based on the Harmonic Sequence The is 0. Think about it. What about As x gets really really huge, what will the reciprocal of x approach? Yup. So we will make use of this concept to evaluate limits as x gets huge.

8. Some rules of limits The great thing about limits is that the limit of something complicated can be done as the limit of all the pieces. Don’t write this down

9. So, to use the harmonic sequence we will multiply top and bottom by where n is the highest overall power you see in the problem. Then evaluate each of the pieces.

10. Examples Find the following Now there is a shortcut “trick” to these problems. WITHOUT TALKING TO ANYONE tonight see if you can figure it out.