To solve problem 54, we need to show that

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

for any $x > 0$.

This is a classical limit that represents the definition of the exponential function $e^x$. Here’s a step-by-step proof:

1. Express the Limit in Exponential Form:

   Recognize that the given limit resembles the form used in the definition of the exponential function $e$. We can rewrite the expression inside the limit:

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

   This can be rewritten by taking the natural logarithm of the limit. Let $L$ be the limit we are trying to find:

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

2. Apply the Natural Logarithm:

   To simplify the exponentiation, take the natural logarithm on both sides:

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

   Using the property of logarithms:

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

   $\ln L = \lim_{n \to \infty} n \ln \left( 1 + \frac{x}{n} \right)$

3. Use the Approximation for Small Values:

   For large $n$, $\frac{x}{n}$ becomes very small. We can use the first-order Taylor expansion of $\ln(1 + u) \approx u$ for $u$ near 0:

   $\ln \left( 1 + \frac{x}{n} \right) \approx \frac{x}{n}$

   Substituting this approximation in:

   $\ln L = \lim_{n \to \infty} n \cdot \frac{x}{n}$

4. Simplify the Expression:

   Simplify the limit:

   $\ln L = \lim_{n \to \infty} x = x$

5. Exponentiate Both Sides:

   To solve for $L$, exponentiate both sides to remove the natural logarithm:

   $L = e^x$

Thus, we have shown that:

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

for any $x > 0$.

Would you like further details or have any questions?

Here are 5 more questions you might want to explore:

1. How do you find the derivative of a product of two functions?
2. What is the chain rule in calculus and how is it applied?
3. Can you solve problem 27 from the provided image?
4. How do you evaluate the integral of a function?
5. What is the fundamental theorem of calculus?

Tip: When dealing with limits, approximations such as Taylor expansions can be very useful for simplifying complex expressions.