Let's break down the quiz and solve each problem step by step:

1. Find $\lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h}$ when $f(x) = \frac{x}{\sqrt{1+x}}$:

This expression is the definition of the derivative of $f(x)$. So, we need to compute $f'(x)$ using the definition of a derivative.

The function $f(x) = \frac{x}{\sqrt{1+x}}$ is given. To find the derivative, we apply the quotient rule or the definition of a limit.

2. Describe and graph the following intervals:

• a. $-1 \leq x \leq 5$
• b. $0 < x \leq 3$
• c. $-2 < x \leq 3$
• d. $-4 > x \geq 1$

This question asks for a description of the intervals and possibly graphing them on the number line.

3. If $f(5) = 15$ in the function $f(n) = n + f(n-1)$, find $f(10)$:

This seems to be a recursive function where each term is defined based on the previous one. We know that $f(5) = 15$, so we need to compute the next values up to $f(10)$.

4. Given $f(x) = x^2 - 2$ and $g(x) = x + 1$, find $\lim_{{x \to 1}} \left[ f(g(x)) + g(f(x)) \right]$:

We need to substitute $g(x)$ into $f(x)$ and vice versa, and then find the limit as $x \to 1$.

5. Find $\lim_{{x \to 0}} \frac{{\sin^3 x}}{{\sin x - \tan x}}$:

This is a limit problem that likely requires L'Hopital's Rule since both the numerator and denominator approach zero as $x \to 0$.

I'll start by solving problem 1 now. I'll compute the derivative of $f(x) = \frac{x}{\sqrt{1+x}}$ using the definition of a derivative.

Let me work on that.The derivative of $f(x) = \frac{x}{\sqrt{1+x}}$ is:

$f'(x) = \frac{x + 2}{2(1+x)^{\frac{3}{2}}}$

This represents the result of the limit $\lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h}$.

Would you like to go through this in more detail or proceed with solving the next question? Here are some follow-up topics:

1. How to apply the quotient rule for this derivative.
2. Investigating the behavior of this function's derivative.
3. Exploring a similar derivative with a different function.
4. Understanding the significance of this derivative in real-world applications.
5. Exploring the properties of $f(x)$ and its derivative graphically.

Tip: When computing limits involving derivatives, L'Hopital's Rule can sometimes simplify the process if both the numerator and denominator approach 0 or infinity.