**(1a)**

*7 points*. Use limit rules to evaluate $\lim_{x \to 9} \frac{\sqrt{x} - 3}{x - 9}$. Show your steps.

**(1b)**

*3 points*. Let $f(x) = \sqrt{x}$, with its natural domain. Does $f'(9)$ exist? Justify very briefly.

**(2a)**

*6 points*. Let $f(x) = \frac{(x - 1)(x - 4)(x - 6)}{5(x - 2)(x - 3)(x - 4)}$. With its natural domain. Find all asymptotes of the graph $f$. (you need not sketch the graph, just indicate the types and locations of the asymptotes.)

**(2b)**

*6 points*. Find $\lim_{x \to 0}x \cos(e^{2/x})$. Justify your answer briefly, using any limit rules or theorems from this course.

**(3)**

*6 points*. Show that there is at least one real number $x$ which satisfies $x^6 = 1 + \sin(x)$.

**(4a.)**Let $t > 0$. How is $\log_t(3)$ defined?

**(4b.)**Let $f(x) = \tan(x)$ with domain $( \frac{\pi}{2}, \pi)$. Does $f$ has an inverse? If so, what are the domain and range of the inverse function?

**(4c.)**If some

*vertical*line intersects a graph at a more than one point, what does it say about the graph?

**(4d.)**Simplify: $\ln(5e\sqrt{x})$, assuming that $x > 0$.

**(4e.)**If the domain of $f$ contains $(-1,1)$, and if $f$ is continuous at $0$, must $f’(0)$ exist?

Either explain in words why it must exist, or give an example of a function for which it does not exist.

**(5a)**

*5 points.*Let $f(x) = x^2$. Find $\delta > 0$ such that $|f(x) - 36| < \frac{1}{1000}$ whenever $|x - 6| < \delta$, and

**show your reasoning**in full detail.

*You need not simplify any numbers which arise, and there is no penalty if your $\delta$ is smaller than necessary.*

**(5b)**

*7 points.*Show, using the precise definition of limit, that $\lim_{x \to \frac{1}{2}} (x - \frac{1}{4x}) = 0$

*(Here "show" means prove".)*

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