Here are the problems from the image:

Question №4

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

Question №5

Find the derivative of the function:
$y = \ln^3{x}$

Solutions

Solution to Question №4:

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

This is a standard limit that results in the base of the natural logarithm $e$:
$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e$

Thus, the answer is:
$\boxed{e}$

Solution to Question №5:

To find the derivative of $y = \ln^3{x}$, we apply the chain rule. Rewrite the function:
$y = (\ln{x})^3$

The derivative is:
$\frac{dy}{dx} = 3 (\ln{x})^2 \cdot \frac{d}{dx}(\ln{x})$

The derivative of $\ln{x}$ is $\frac{1}{x}$. Therefore:
$\frac{dy}{dx} = 3 (\ln{x})^2 \cdot \frac{1}{x}$

Simplifying:
$\frac{dy}{dx} = \frac{3 (\ln{x})^2}{x}$

Thus, the derivative is:
$\boxed{\frac{3 (\ln{x})^2}{x}}$

Do you need further clarification on any step? Here are 5 related questions for practice:

1. Evaluate $\lim_{x \to \infty} \left(1 + \frac{k}{x}\right)^x$ where $k$ is a constant.
2. Differentiate $y = \ln^4{x}$.
3. Prove that $\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^{x+1} = e$.
4. Find the derivative of $y = x^2 \ln{x}$.
5. Find the second derivative of $y = \ln{x}$.

Tip:

The limit $\left(1 + \frac{1}{x}\right)^x$ is a fundamental result that defines the constant $e$. Understanding it is key for exponential and logarithmic growth analysis.