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 $\mathbb{R}$: