To evaluate the limit

$\lim_{x \to \infty} \left( 1 + \frac{1}{x} \right)^x$

using L'Hôpital's Rule, we need to rewrite the expression in a form suitable for the rule. Here's the step-by-step process:

Step 1: Rewrite the expression

Let: $y = \left( 1 + \frac{1}{x} \right)^x$

Take the natural logarithm on both sides: $\ln y = x \ln\left( 1 + \frac{1}{x} \right)$

Now, the problem becomes finding the limit of $\ln y$: $\lim_{x \to \infty} x \ln\left( 1 + \frac{1}{x} \right)$

Step 2: Simplify the expression

The expression $x \ln\left( 1 + \frac{1}{x} \right)$ can be written as: $\ln y = \frac{\ln\left( 1 + \frac{1}{x} \right)}{\frac{1}{x}}$

As $x \to \infty$, this takes the indeterminate form $\frac{0}{0}$. We can apply L'Hôpital's Rule.

Step 3: Apply L'Hôpital's Rule

Differentiate the numerator and denominator with respect to $x$:

• The derivative of the numerator $\ln\left( 1 + \frac{1}{x} \right)$ is: $\frac{1}{1 + \frac{1}{x}} \cdot \left(-\frac{1}{x^2}\right) = -\frac{1}{x^2\left( 1 + \frac{1}{x} \right)}$
• The derivative of the denominator $\frac{1}{x}$ is: $-\frac{1}{x^2}$

Now, the limit becomes:

= \lim_{x \to \infty} \frac{1}{1 + \frac{1}{x}}$$ As $$x \to \infty$$, $$\frac{1}{x} \to 0$$, so: $$\lim_{x \to \infty} \frac{1}{1 + \frac{1}{x}} = 1$$ ### Step 4: Exponentiate the result Since $$\ln y = 1$$, we have: $$y = e^1 = e$$ Thus: $$\lim_{x \to \infty} \left( 1 + \frac{1}{x} \right)^x = e$$ ### Final Answer: $$\lim_{x \to \infty} \left( 1 + \frac{1}{x} \right)^x = e$$ --- Would you like a more detailed explanation of any step, or help with similar problems? --- ### Follow-up Questions: 1. What is the importance of exponential limits like $$\lim_{x \to \infty} \left( 1 + \frac{1}{x} \right)^x$$? 2. How can we interpret this limit in real-world contexts, such as continuous compound interest? 3. Can the same approach be used for $$\lim_{x \to \infty} \left( 1 + \frac{k}{x} \right)^x$$? 4. How does the natural exponential function arise in calculus from limits? 5. Are there other cases where logarithms simplify exponential limits? --- **Tip:** Always check the form of the limit (e.g., $$0/0$$, $$\infty/\infty$$) before applying L'Hôpital's Rule to avoid unnecessary steps.