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Take It To the Limit Limits of Functions

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A limit example Suppose the frog can only hop half-way to the stream on each hop. Would he ever get to the water?

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Reaching the Limit If half the remaining distance is covered with each jump, theoretically the frog will never reach the stream. The stream represents the limit. Limits in mathematics are similar.

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**Definition of a limit (informal definition)**

If f(x) becomes arbitrarily close to a single number L as x approaches c from either side, the limit of f(x), as x approaches c, is L.

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An Example Suppose you are asked to find the limit of the function:

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**Solution to the Example**

Replace the variable (x) with the number –1. This would result in the following:

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**Example Using Excel Spreadsheet**

Given the limit problem: Notice what happens when you evaluate numbers closer and closer to 2

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**A graphical illustration**

Given the limit to evaluate Graph the function: f(x) = 2x-5

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Table view of function Notice at the value x=2, the f(x) or y value is -1

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Conclusion Limits of functions can be found by evaluating the function for the value the variable is approaching, provided the function is defined at that value.

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Copyright © Cengage Learning. All rights reserved. 3 Applications of Differentiation.

Copyright © Cengage Learning. All rights reserved. 3 Applications of Differentiation.

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