Find the domain of the function. Express your answer in interval notaion. (10 pts.)

Find the value of a that makes the function
continuous at $x=2$. (20 pts.)

Find the derivative $f'(x)$.
$f(x) = \frac{2x - 1}{x^2 + 1}$

A particles position in $cm$ after $t$ seconds is given by the function
Find the velocity of the particle at time $t=0$.

Find the limits if they exist. Otherwise, write $+ \infty \; \text{DNE}, -\infty \; \text{DNE}$, or just $\text{DNE}$.

(a) $\lim_{x \to \infty} \frac{3x^2 - 3x + 3}{2x^3 + 10x}$

(a) $\lim_{x \to \infty} \frac{3x^2 - 3x + 3}{2x^3 + 10x}$

(b) $\lim_{x \to 5^{+}} \frac{4x - 5}{30 - 6x}$

(c) $\lim_{x \to -1} \frac{2x \cos(\pi x)}{x^2 + 1}$

(d) $\lim_{x \to 3} \frac{2x - 6}{9 - x^2}$

6. Use the figure below to answer the questions. The figure depicts the graph of the function y=f(x). (4 pts. each)

(d) Give an estimate for $f'(4)$ and sketch the tangent line at $x = $ in the figure above.

(e) For what values of $x$ is $f(x)$ continuous? Give your answer in interval notation.

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