Calculus: Finding Limits with Limit Laws

The limit laws are sort of like the properties of logarithms or the trigonometric identities you've used in the past. They are laws (that have been proven to be true) that make complicated calculations easier. I've typed up the basic limit laws into a Google Document that you can view by following this link. You must have the linke to view this document. If you save the link/bookmark it and then find it is not working, come back to this page and get the link again. It's possible I may have to refresh the link from time to time, so if you can't get it to work, leave a comment on this post and I will try to fix it!

The image is too large to put in this post for reference, so open the link above and have it in a separate window as you go through this topic.

Along with the limit laws I've given you, you'll need a couple of other theorems.

The first is simple. If a function f(x) is less than a function g(x) when x is near a, then the limit of f(x) as x approaches a is less than the limit of g(x) as x approaches a. This makes sense. If one function is less than another at a particular value, then its limit will be less than the limit of the other at that value.

The "Squeeze Theorem" is similar, but uses three functions.

This also makes sense. If a function (here, g(x)) is "squeezed" between two other functions near x = a, and if the two outside functions have limits of L, then the function squeezed between (g(x)) must have the same limit.

Let's practice using the limit laws and the squeeze theorem.

Here, I've detailed each step of the process. The first step, I used property 3. In the second, I used the property of polynomials and property 3. Next, I used property 6, then property 10, then the polynomial property again to arrive at my final answer.

And here is a problem that uses the squeeze theorem.

Here, we are told that f(x) is between these other two functions whenever x is greater than or equal to 0. We are asked to find the limit of f(x). So, by the squeeze theorem, we know that the limit of f(x) is equal to the limit of the other two functions. I found the limit of each of the other two functions (you really only need one, but it is good to check that these two limits are equal), and then I know that the limit of f(x) is the same.

Sometimes, you will be asked to find limits regarding piecewise functions, like below. Questions are in bold, answers in regular font.

Notice that for handed-limits as in part (i), (iv), and (v), in a piecewise function, you need to know which piece of the function you are considering. In part (i), we are approaching 1 from the positive side, so we use the piece of the function where x > 1, which is the third portion. In part (iv), we use the first portion because we want to approach -1 from the left side, that is, where x < -1. Also notice that the limit in part (vi) doesn't exist because the right- and left-handed limits (parts (iv) and (v)) are not equal.

I hope this post has helped you. Limits are not difficult--you just need to know the limit laws. Practice limits. If you are struggling to find an answer using the limit laws, graph the function and see if you can get an estimate. This isn't a sufficient answer on homework or a test, but maybe can help you see where you are going.

Also, if a limit does not exist, you cannot use the limit laws for it! You will need to graph the function and estimate a limit or use the squeeze theorem. Find two functions, one that is less than the questioned function and one that is greater at the point of interest, and then find the limits of those. Note that polynomials are the easiest limits to find, so try to find polynomials when using the squeeze theorem.