https://file.aiquickdraw.com/mathbot/file/1730428987896f1qaf9um.jpg

Let's go through each problem in the image.

### 1. Tangent Line
Find the equation of the line tangent to \( f(x) = x^2 + 3x \) at \( x = 2 \).

To find the tangent line equation:
1. Calculate \( f'(x) \) to find the slope of the tangent at \( x = 2 \).
2. Evaluate \( f(2) \) and \( f'(2) \) to form the tangent line equation using \( y - f(2) = f'(2)(x - 2) \).

### 2. Limits
Evaluate each of the following limits:
   - (a) \( \lim_{x \to 8} \frac{3x - 2 + \sqrt{x + 1}}{x - 8} \)
   - (b) \( \lim_{x \to 1} \frac{x^3 - 1}{x - 1} \)
   - (c) \( \lim_{x \to 0} \frac{|x|}{x} \)
   - (d) \( \lim_{x \to 2} \frac{x|x - 2|}{x - 2} \)

These involve different techniques:
   - Simplifying expressions.
   - L'Hôpital's Rule.
   - Understanding limits from the left and right for absolute values.

### 3. Calculus Questions
   - (a) Differentiate \( f(x) = \frac{3x - 8}{\sqrt{x - 3}} \) and find \( f'(-1) \).
   - (b) For \( y = x(x^3 + 1)^5 \), find \( \frac{dy}{dx} \).
   - (c) Calculate the sum for \( 1 + x + x^2 + x^3 + \dots + x^n \), \( x \neq 1 \), which is a geometric series.
   - (d) Given functions \( f \) and \( g \) with values for \( g'(2) \) and \( g(2) \), find \( h'(2) \) for \( h(x) = f(g(x)) \) using the chain rule.

### 4. Continuity of Piecewise Function
Determine the value of \( c \) so that \( f(x) \) is continuous on \( \mathbb{R} \):
\[
f(x) = 
\begin{cases} 
cx^2 - 3, & x \leq 2 \\
cx + 2, & x > 2 
\end{cases}
\]
To make \( f(x) \) continuous, equate the left and right limits as \( x \to 2 \).

### 5. Greatest Integer Function and Limits
Given \( f(x) = 
\begin{cases} 
x \lfloor x \rfloor, & x < 2 \\ 
2x - 2, & x \geq 2 
\end{cases} \)
determine:
   - (a) \( \lim_{x \to 2} f(x) \) and whether it exists.
   - (b) \( f'(2) \) and whether it exists.

### 6. Chain Rule Application
If \( f(1 - x^3) = 5x^2 \), find \( f'(9) \).

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### Further Questions:
1. How do you find the tangent line to a curve at a specific point?
2. What are common techniques to evaluate indeterminate limits?
3. What approach should you take to differentiate functions involving products and compositions?
4. How do you ensure continuity in a piecewise function?
5. What is the greatest integer function, and how does it affect limit calculations?

**Tip:** When dealing with limits that result in indeterminate forms like \( \frac{0}{0} \), try using L'Hôpital's Rule or algebraic simplification to resolve the expression.

What is the solution to a set of calculus questions, including tangent line equation, limits, derivatives, continuity of piecewise functions, greatest integer function limits, and chain rule application?

Solve a calculus problem set covering tangent line equations, limits of functions, continuity of piecewise functions, and application of the chain rule. This guide includes techniques for evaluating indeterminate forms, differentiating composite functions, and ensuring continuity in piecewise definitions. Suitable for college calculus students.

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