https://file.aiquickdraw.com/mathbot/file/1726091644021eh3ke9y9.jpg

The image provided contains two limit problems to be evaluated. Let's analyze and solve each one:

### Problem (a)
\[
\lim_{x \to \infty} \frac{2}{e^x - 7}
\]
As \(x \to \infty\), the exponential function \(e^x\) grows very rapidly. Therefore, the term \(e^x - 7\) will also grow very large because subtracting 7 from a very large number will not change its magnitude significantly.

Thus, we can approximate:
\[
e^x - 7 \approx e^x \quad \text{as} \quad x \to \infty
\]
So, the limit becomes:
\[
\lim_{x \to \infty} \frac{2}{e^x} = 0
\]
Therefore, the solution to (a) is:
\[
\boxed{0}
\]

### Problem (b)
\[
\lim_{x \to -\infty} \frac{2}{e^x - 7}
\]
As \(x \to -\infty\), the exponential function \(e^x\) approaches 0, because \(e^x\) decays rapidly as \(x\) becomes more negative. Therefore, \(e^x - 7\) approaches:
\[
e^x - 7 \approx -7 \quad \text{as} \quad x \to -\infty
\]
So the limit becomes:
\[
\lim_{x \to -\infty} \frac{2}{-7} = -\frac{2}{7}
\]
Thus, the solution to (b) is:
\[
\boxed{-\frac{2}{7}}
\]

### Summary of Results:
- (a) \(\lim_{x \to \infty} \frac{2}{e^x - 7} = 0\)
- (b) \(\lim_{x \to -\infty} \frac{2}{e^x - 7} = -\frac{2}{7}\)

Would you like a detailed explanation on any of the steps?

Here are 5 related questions you might find helpful:
1. What happens if the function inside the limit is polynomial instead of exponential?
2. How does the behavior of exponential functions compare to logarithmic functions as \(x \to \infty\)?
3. What are the properties of the exponential function \(e^x\) for large negative \(x\)?
4. How can L'Hopital's Rule be used for evaluating limits involving indeterminate forms?
5. How do horizontal asymptotes relate to limits as \(x\) approaches infinity?

**Tip:** When solving limits involving exponentials, always remember the rapid growth rate of \(e^x\) as \(x \to \infty\). It helps simplify complex expressions.

Evaluate the following limits: (a) lim(x→∞) 2 / (e^x - 7), (b) lim(x→-∞) 2 / (e^x - 7).

Solve the limits lim(x→∞) 2 / (e^x - 7) = 0 and lim(x→-∞) 2 / (e^x - 7) = -2/7 with a step-by-step explanation of exponential function behavior. Suitable for college-level 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