Find a Function f(x) with Limit at Infinity but no Derivative Limit

Thread starter AlexHall
Start date Nov 13, 2008
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Derivative Function Infinity Limit Limit at infinity

In summary, a limit at infinity is the value that a function approaches as the input variable approaches positive or negative infinity. This value is denoted as lim f(x) as x approaches infinity. A function can have a limit at infinity but no derivative limit, meaning that it approaches a certain value at infinity but the slope does not have a finite limit. To find such a function, you can look for functions with vertical asymptotes at infinity or manipulate existing functions. The difference between a limit at infinity and a limit at a specific value is that the former determines the overall behavior of a function while the latter determines the behavior near a specific value. It is important to understand functions with limits at infinity but no derivative limit as they provide insight into complex

Nov 13, 2008

AlexHall

Hello

I need to give an example of a function f(x) such that limf(x), as x goes to infinity, exists but the limf'(x) does not. Any help?

Thx

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Nov 13, 2008

Yalldoor

Try exploring negative exponentials.

1. What is the definition of a limit at infinity?

A limit at infinity is the value that a function approaches as the input variable approaches positive or negative infinity. It is denoted as lim f(x) as x approaches infinity.

2. Can a function have a limit at infinity but no derivative limit?

Yes, a function can have a limit at infinity but no derivative limit. This means that the function approaches a certain value at infinity, but the slope of the function does not have a finite limit at infinity.

3. How can I find a function with a limit at infinity but no derivative limit?

To find a function with a limit at infinity but no derivative limit, you can start by looking for functions that have vertical asymptotes at infinity, such as rational functions with a higher degree in the denominator than the numerator. You can also manipulate existing functions by adding or subtracting terms that approach infinity as the input variable approaches infinity.

4. What is the difference between a limit at infinity and a limit at a specific value?

A limit at infinity is the value that a function approaches as the input variable approaches positive or negative infinity, while a limit at a specific value is the value that a function approaches as the input variable approaches a specific value. A limit at infinity is used to determine the overall behavior of a function, while a limit at a specific value is used to determine the behavior near that specific value.

5. Why is it important to understand functions with limits at infinity but no derivative limit?

Understanding functions with limits at infinity but no derivative limit is important because it allows us to analyze the behavior of functions that do not have a well-defined slope at infinity. These functions often have interesting and unique characteristics that can help us understand more complex mathematical concepts. It is also important in real-world applications, such as in physics and economics, where functions with limits at infinity but no derivative limit can model certain behaviors and phenomena.