https://r2.erweima.ai/mathbot/image/1725987054692zm3i0xxn.jpg

We are tasked with evaluating the following limit as \( h \to 0 \):

\[
\lim_{h \to 0} \frac{\sqrt[5]{5^5 + h} - 5}{h}
\]

This expression is an indeterminate form of type \( \frac{0}{0} \) when \( h = 0 \). We will need to apply L'Hôpital's Rule, which is used to resolve limits of indeterminate forms, or alternatively apply a first-order approximation using a Taylor series or binomial expansion.

### Solution using L'Hôpital's Rule:
1. **Differentiate the numerator**: The numerator is \( \sqrt[5]{5^5 + h} - 5 \), so we need to find its derivative with respect to \( h \).
   
   Using the chain rule, the derivative of \( \sqrt[5]{x} = x^{1/5} \) is \( \frac{1}{5} x^{-4/5} \), and applying this to \( x = 5^5 + h \), we get:
   \[
   \frac{d}{dh} \left( \sqrt[5]{5^5 + h} \right) = \frac{1}{5} (5^5 + h)^{-4/5}
   \]

2. **Differentiate the denominator**: The derivative of \( h \) with respect to \( h \) is just \( 1 \).

After differentiation, we evaluate the limit by substituting \( h = 0 \).

Let me calculate the result.The limit evaluates to:

\[
\lim_{h \to 0} \frac{\sqrt[5]{5^5 + h} - 5}{h} = 0.00032
\]

Would you like more details or have any questions about the solution?

Here are some related questions to further explore this topic:
1. How is L'Hôpital's Rule applied to other types of indeterminate forms?
2. Can we approximate this limit using a binomial expansion?
3. What happens if we increase the exponent in the root function, like a 7th root instead of a 5th?
4. How does the chain rule work when differentiating compositions of functions like this?
5. What are other methods to evaluate limits that result in indeterminate forms?

**Tip:** Always verify if L'Hôpital’s Rule applies by confirming the indeterminate form, such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), before applying it.

Determine lim as h -> 0 of (5th root(5^5 + h) - 5)/h

Learn how to solve the limit problem lim as h -> 0 of (5th root(5^5 + h) - 5)/h using L'Hôpital's Rule. This problem involves derivatives and the chain rule. Suitable for advanced high school or early college students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation