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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 \( x \to 0 \).

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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 \( \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h} \).

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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.

Find limit, describe and graph intervals, solve recursive function, and evaluate given limits.

Explore advanced calculus problems including limits, recursive functions, trigonometric limits, and graphing intervals. Detailed step-by-step solutions to evaluate derivatives, composite functions, and recursive sequences.

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