What is the value of  \lim_{h\to0} \frac{\left(16+h\right)^{\frac{1}{4}}-2}{h}  ?       

We need to compute the value of the limit:

\[
\lim_{h \to 0} \frac{(16 + h)^{\frac{1}{4}} - 2}{h}
\]

This limit can be solved using the **derivative definition of a function**. Notice that this expression resembles the definition of the derivative for a function at a specific point. Let's rewrite this limit in terms of derivatives.

### Step 1: Recognize the form
Let \( f(x) = x^{\frac{1}{4}} \). The expression:

\[
\frac{(16 + h)^{\frac{1}{4}} - 2}{h}
\]

is similar to the definition of the derivative of \( f(x) \) at \( x = 16 \):

\[
f'(16) = \lim_{h \to 0} \frac{f(16 + h) - f(16)}{h}
\]

Here, \( f(x) = x^{\frac{1}{4}} \), so:

\[
f(16) = 16^{\frac{1}{4}} = 2
\]

Therefore, the given limit is just the derivative of \( f(x) = x^{\frac{1}{4}} \) evaluated at \( x = 16 \).

### Step 2: Compute the derivative
Now, let's compute the derivative of \( f(x) = x^{\frac{1}{4}} \):

\[
f'(x) = \frac{d}{dx}\left(x^{\frac{1}{4}}\right) = \frac{1}{4} x^{-\frac{3}{4}}
\]

Evaluating this derivative at \( x = 16 \):

\[
f'(16) = \frac{1}{4} \cdot 16^{-\frac{3}{4}} = \frac{1}{4} \cdot \frac{1}{16^{\frac{3}{4}}} = \frac{1}{4} \cdot \frac{1}{8} = \frac{1}{32}
\]

### Step 3: Conclusion
Thus, the value of the limit is:

\[
\boxed{\frac{1}{32}}
\]

Would you like more details or have any questions? 

Here are five related questions for further practice:

1. What is the derivative of \( f(x) = x^{\frac{1}{2}} \) at \( x = 25 \)?
2. How would you evaluate \( \lim_{h\to0} \frac{(25+h)^{\frac{1}{2}}-5}{h} \)?
3. What is the general form of the derivative for \( f(x) = x^{n} \), where \( n \) is any rational number?
4. What is the geometric interpretation of the derivative of a function?
5. How would you apply the chain rule to compute the derivative of \( f(x) = (3x+2)^{\frac{1}{4}} \)?

Tip: Derivatives are powerful for calculating rates of change, especially for functions involving powers and roots!

What is the value of \lim_{h\to0} \frac{\left(16+h\right)^{\frac{1}{4}}-2}{h}?

Learn how to evaluate the limit \lim_{h\to0} \frac{(16+h)^{\frac{1}{4}} - 2}{h} by using the derivative of the function x^(1/4). This solution covers key calculus concepts, including limits and derivatives. Suitable for students in Grades 11-12.

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