Help With Limit: a=16, f=${x}^{.25}$

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In summary, the conversation discusses a limit problem and asks for the value of a and f. The answer for part (a) is 16 and the function for part (b) is explained as f(x) = x^(0.25) = √x, with f(16) = 2 and f(16+h) = √(16+h). The person expresses gratitude for the clarification.
  • #1
lastochka
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Hello, I have this homework questions with answers. I got part (a) a=16, but part (b) f=${x}^{.25}$ I don't understand...
Here is the problem:
This limit represents the derivative of some function f at some number a. State this a and f
$\lim_{{h}\to{0}}$$\frac{\sqrt[4]{16+h}-2}{h}$
Part a) is obvious the answer is 16 just by looking at it.
Please, help me to understand why in part b) f=${x}^{.25}$
Thank you in advance!
 
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  • #2
Hint:
$$f(x) = x^{0.25} = \sqrt[4]{x}$$
Now:
$$f(16) = 16^{0.25} = \sqrt[4]{16} = 2$$
And:
$$f(16 + h) = (16 + h)^{0.25} = \sqrt[4]{16 + h}$$
Does that make it clearer?
 
  • #3
Bacterius said:
Hint:
$$f(x) = x^{0.25} = \sqrt[4]{x}$$
Now:
$$f(16) = 16^{0.25} = \sqrt[4]{16} = 2$$
And:
$$f(16 + h) = (16 + h)^{0.25} = \sqrt[4]{16 + h}$$
Does that make it clearer?

Why I didn't see that before! (Giggle)
Thank you so much for your help!
 
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1. What is the definition of a limit?

A limit is a mathematical concept that represents the value that a function approaches as the input approaches a certain value. It is denoted by the symbol "lim" and is often used in calculus to understand the behavior of functions.

2. How do you find the limit of a function?

To find the limit of a function, you can either use algebraic methods such as factoring and simplifying, or you can use graphical methods such as plotting the function or using a graphing calculator. You can also use the limit laws and rules to evaluate the limit of a more complicated function.

3. What is the limit of the given function, a=16, f=${x}^{.25}$?

The limit of the function a=16, f=${x}^{.25}$ is 2. This can be calculated by substituting the given value of a=16 into the function and evaluating the result.

4. What is the significance of finding the limit of a function?

Finding the limit of a function helps us to understand the behavior of the function as the input approaches a certain value. It can also help us to determine the continuity of a function and identify any discontinuities or asymptotes. Limits are also important in calculus, as they are used to calculate derivatives and integrals.

5. Can limits be used to solve real-world problems?

Yes, limits can be used to solve real-world problems, particularly in physics, engineering, and economics. For example, limits can be used to calculate velocities and accelerations, optimize production processes, and determine optimal levels of production. They can also be used to model natural phenomena and predict future events.

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