We are tasked with evaluating the limit:

\[
\lim_{x \to +\infty} \frac{e^x}{\left( 1 + \frac{1}{x} \right)^{x^2}}.
\]

### Step-by-step approach:

1. **Simplify the denominator**:
   - The expression in the denominator is \( \left( 1 + \frac{1}{x} \right)^{x^2} \).
   - We know that as \( x \to +\infty \), \( \left( 1 + \frac{1}{x} \right)^x \to e \) (this is a well-known limit).
   
   Let’s analyze \( \left( 1 + \frac{1}{x} \right)^{x^2} \). We can rewrite this as:
   \[
   \left( 1 + \frac{1}{x} \right)^{x^2} = \left[ \left( 1 + \frac{1}{x} \right)^x \right]^x.
   \]

   Since \( \left( 1 + \frac{1}{x} \right)^x \to e \) as \( x \to +\infty \), the whole expression tends to \( e^x \).

2. **Take the limit of the whole expression**:
   Now, our limit becomes:
   \[
   \lim_{x \to +\infty} \frac{e^x}{e^x} = 1.
   \]

Thus, the limit is:

\[
\boxed{1}.
\]

Would you like more details or explanations on any step?

---

Here are 5 follow-up questions for further exploration:

1. How does the limit of \( \left( 1 + \frac{1}{x} \right)^x \) as \( x \to +\infty \) lead to the constant \( e \)?
2. What are the implications of L'Hopital's rule for evaluating indeterminate forms of limits?
3. How does the exponential function \( e^x \) behave as \( x \to +\infty \)?
4. How would the limit change if the denominator exponent were \( x^3 \) instead of \( x^2 \)?
5. How do other small terms like \( \frac{1}{x} \) affect the behavior of functions as \( x \to \infty \)?

**Tip:** When dealing with complex limits, rewriting terms in a simpler form or using standard limits can greatly simplify the process.

Explore the step-by-step solution for evaluating the limit of e^x divided by (1 + 1/x) raised to the power of x^2 as x approaches infinity. This calculus problem highlights key concepts such as exponential limits and L'Hopital's Rule, suitable for advanced high school and undergraduate 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