https://file.aiquickdraw.com/mathbot/file/1725505844832mngsyfdg.jpg

# Since the user has uploaded an image, I will first use OCR to extract the mathematical expression from the image for further processing.
import pytesseract
from PIL import Image

# Load the image from the file path
img_path = "/mnt/data/1725505844832mngsyfdg.jpg"
img = Image.open(img_path)

# Use pytesseract to extract text
extracted_text = pytesseract.image_to_string(img)
extracted_textThe image text extraction didn't work correctly. However, I can directly work with the image content you've provided. The limit in question is:

\[
\lim_{{h \to 0}} \frac{7e^x - 7e^{x+h}}{4h}
\]

Let's solve this limit step by step.

### Step 1: Rewrite the expression
We know that \( e^{x+h} = e^x \cdot e^h \), so we can rewrite the limit as:

\[
\lim_{{h \to 0}} \frac{7e^x - 7e^x \cdot e^h}{4h}
\]

### Step 2: Factor out common terms
Factor out \( 7e^x \) from the numerator:

\[
\lim_{{h \to 0}} \frac{7e^x \left(1 - e^h\right)}{4h}
\]

### Step 3: Use the series expansion for \( e^h \)
For small \( h \), we know the series expansion for \( e^h \) is:

\[
e^h \approx 1 + h + \frac{h^2}{2!} + \cdots
\]

Substitute this into the limit expression:

\[
\lim_{{h \to 0}} \frac{7e^x \left(1 - \left(1 + h + \frac{h^2}{2!} + \cdots \right)\right)}{4h}
\]

Simplify the numerator:

\[
1 - \left(1 + h + \frac{h^2}{2!} + \cdots \right) = -h - \frac{h^2}{2!} - \cdots
\]

Thus, the expression becomes:

\[
\lim_{{h \to 0}} \frac{7e^x \left(-h - \frac{h^2}{2!} - \cdots \right)}{4h}
\]

### Step 4: Simplify and evaluate the limit
Now, divide by \( h \):

\[
\lim_{{h \to 0}} \frac{7e^x \left(-h\right)}{4h} = \lim_{{h \to 0}} \frac{-7e^x}{4}
\]

The higher-order terms vanish as \( h \to 0 \), leaving us with:

\[
\boxed{\frac{-7e^x}{4}}
\]

Thus, the value of the limit is \( \frac{-7e^x}{4} \).

Let me know if you'd like any further explanation or clarification!

Here are 5 questions you might want to explore next:
1. How do higher-order terms affect limits involving exponential functions?
2. What is the general form of the derivative of \( e^x \)?
3. How would the solution change if the limit involved \( \cos(x) \) instead of \( e^x \)?
4. What does the series expansion for \( e^h \) tell us about small values of \( h \)?
5. How can L'Hopital's Rule be used to solve limits like this one?

**Tip**: When dealing with limits involving exponential functions, Taylor series expansions are a powerful tool for simplifying expressions around small values of the variable.

Evaluate the limit lim_{h→0} (7e^x - 7e^{x+h}) / 4h.

Explore how to evaluate the limit of the expression (7e^x - 7e^{x+h}) / 4h as h approaches 0. This problem utilizes exponential functions, Taylor series expansion, and basic calculus concepts. Suitable for university-level 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