What makes a limit indeterminate

This will help us when it comes time to take some derivatives. However, as we saw in the last example we need to be careful with how we do that on occasion. Sometimes we can use either quotient and in other cases only one will work. So, what did this have to do with our limit? Well first notice that,. Notes Quick Nav Download. You appear to be on a device with a "narrow" screen width i. Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device should be able to scroll to see them and some of the menu items will be cut off due to the narrow screen width.

Example 1 Example 1 Evaluate each of the following limits. Example 2 Evaluate the following limit. Example 3 Evaluate the following limit. Example 4 Evaluate the following limit. For example, if. Before we continue, we need to draw attention to a notation that we have been using when calculating limits.

We now encourage the reader to investigate each one of the terms shown in Table 5. An undefined expression involving some operation between two quantities is called a determinate form if it evaluates to a single number value or infinity. An undefined expression involving some operation between two quantities is called an indeterminate form if it does not evaluate to a single number value or infinity.

We will inspect multiplication more closely. This is like two ends of a rope being tugged and we do not know which side is going to win. We leave the remaining terms up to the reader to investigate and simply present the determinate and indeterminate forms of the expressions from Table 5.

We are now in a position to introduce one more technique for trying to evaluate a limit. This theorem is somewhat difficult to prove, in part because it incorporates so many different possibilities, so we will not prove it here. For this reason one had to refrain from defining any value at the indeterminate forms, so evaluation will come to a halt when it finds one on the way.

IMO, the simplest explanation involves Taylor series and maybe Laurent series, although we can use differential approximation to get the flavor of what's going on. Exercise : prove that the above is true.

The answer is that their limits are not indeterminate. In fact, their limits are well-defined. This is because those limits are also well-defined i. L'Hopital's Rule only applies when the limit cannot be defined. Hence it is an indeterminate quantity.

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