The problem presented is:

$\lim_{x \to \infty} \left( 1 + \frac{4}{x} \right)^x$

To solve this, let’s proceed with the following steps:

Step 1: Recognize the Form

This limit is of the form $(1 + \frac{k}{x})^x$ as $x \to \infty$, which is similar to the well-known limit that leads to an exponential function: $\lim_{x \to \infty} \left( 1 + \frac{k}{x} \right)^x = e^k$ where $k$ is a constant.

In this case, $k = 4$.

Step 2: Apply the Exponential Limit Property

Using the formula above, we have: $\lim_{x \to \infty} \left( 1 + \frac{4}{x} \right)^x = e^4$

Conclusion

The value of the limit is: $e^4$

Would you like more details on this solution or have any other questions?

Here are some related questions to deepen understanding:

1. How can we derive the formula $\lim_{x \to \infty} \left( 1 + \frac{k}{x} \right)^x = e^k$?
2. What is the value of $e^4$ approximately?
3. How would the solution change if the expression were $\lim_{x \to \infty} \left( 1 + \frac{4}{x^2} \right)^x$?
4. Can you use L'Hôpital's Rule to evaluate limits of this form?
5. How does the concept of limits relate to exponential growth?

Tip: Recognizing common limit forms can greatly speed up solving limits approaching infinity.