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This section includes both 1 and 2 sided limits. You will be given a graph or a function and will be asked to determine the limit at a number of points.
The list of danger zones below shows places where you must be very careful to test the limit.
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.
See the example below:
The answers to the question above are 2.1) 2 2.2) -13, Undefined, Undefined
Find more examples (with answers) here:
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