3. Asymptotes and Limits at 0 and infinity

This section is difficult and requires analysis of the function to figure out more challenging limits.

End behavior (for limits going to infinity or negative infinity):

REMEMBER TO USE HIGHEST POWER TERMS FOR THIS

Big/small -> Infinity or negative infinity

Same power -> Ratio of the leading terms

Small/Big -> Zero

Behavior for limits at zero

REMEMBER TO USE THE LOWEST POWER TERMS FOR THIS

Small/Big = Infinity or negative infinity

Same Power -> Ratio of the smallest power terms

Big/small -> Zero

Danger zones to check for:

1. Holes. If you get a zero/zero situation, you could have a hole and you need to try to find it as the limit MAY exist. To find a hole, factor the top and the bottom and look for terms that will cancel out. If you can cancel a term out at x value of the limit you are evaluating, you have found a hole and if you plug in the limit it may now yield an answer. Practice factoring rational expressions here:

2. Negatives in square roots. If you plug in your limit and you get a negative in a square root, there is no real answer to the problem. If you get zero in the square root, it is important to check whether the function exists on both sides. For example, the square root of x starts at 0 so its limit at 0 is undefined because it only exists from the positive side.

3. Negatives or zeroes in logarithms. The properties of logarithms are the same as square roots with limits with the exception that you cannot take the logarithm of zero.

4. The intersections of graphs in piece-wise functions: In piece-wise functions, the added layer of complexity comes from the intersections between the different parts of the piece-wise functions. In the problem below, to evaluate the limit at x=0, you must plug x=0 into both functions on either side of 0. From the left side you get -13 and from the right side you get 0 meaning the limit does not exist at that point. If they had matched in value, the limit would exist.

5. If a function does not exist on one of the sides of your limit point:

Example problem: