https://file.aiquickdraw.com/mathbot/file/1726444999670hxjnagni.jpg

The problem in the image asks you to estimate the limit:

\[
\lim_{h \to 0} \frac{5^h - 1}{h}
\]

The approach involves substituting smaller and smaller values of \( h \) to approximate the limit. Let’s start by testing a few values for \( h \):

- When \( h = 0.1 \)
- When \( h = 0.01 \)
- When \( h = 0.001 \)
- And so on, getting smaller values closer to 0.

I'll calculate these values to estimate the limit.By substituting smaller values of \( h \), here are the results:

- For \( h = 0.1 \), \( \frac{5^h - 1}{h} \approx 1.75 \)
- For \( h = 0.01 \), \( \frac{5^h - 1}{h} \approx 1.62 \)
- For \( h = 0.001 \), \( \frac{5^h - 1}{h} \approx 1.61 \)
- For \( h = 0.0001 \), \( \frac{5^h - 1}{h} \approx 1.61 \)
- For \( h = 0.00001 \), \( \frac{5^h - 1}{h} \approx 1.61 \)

As \( h \) gets smaller, the values approach approximately \( 1.61 \). Thus, the limit seems to be around **1.6** when rounded to one decimal place.

Would you like more details or have any further questions?

Here are some related questions:
1. What is the general process to solve limits involving exponential functions?
2. Can limits always be calculated by substitution for small values?
3. How would this limit change if the base of the exponent were different?
4. What does the derivative of an exponential function tell us about limits like this one?
5. How would you apply L'Hopital's rule to evaluate this limit?

**Tip:** For limits involving exponential growth, using properties of derivatives can simplify the process.

Estimate the limit by substituting smaller and smaller values of h. Round your estimate to one decimal place: lim (h -> 0) (5^h - 1) / h

Learn how to estimate the limit lim (h -> 0) (5^h - 1) / h by substituting smaller and smaller values of h. Explore the exponential function properties and get a step-by-step solution. Suitable for calculus 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