Limits and Continuity Practice Exam A

A graph approaches $4$ from both sides at $x=2$, but the filled point is $(2,7)$. State $f(2)$ and the limit. Explain the distinction.

State all three continuity conditions at $x=a$.

Explain why $0/0$ is indeterminate rather than equal to zero.

State what the Intermediate Value Theorem guarantees and what it does not guarantee.

$\displaystyle\lim_{x\to-2}(2x^3-x+1)$

$\displaystyle\lim_{x\to3}\frac{x^2-9}{x-3}$

$\displaystyle\lim_{x\to0}\frac{\sqrt{x+9}-3}{x}$

$\displaystyle\lim_{x\to2}\frac{\frac1x-\frac12}{x-2}$

$\displaystyle\lim_{x\to0}\frac{\sin(7x)}{2x}$

$\displaystyle\lim_{x\to0}x^2\cos(1/x)$

$\displaystyle\lim_{x\to1}\frac{|x-1|}{x-1}$

Find both one-sided limits of $\dfrac{x+1}{x-2}$ at $x=2$.

$\displaystyle\lim_{x\to-\infty}\frac{5x^2+1}{2x^2-3x}$

$\displaystyle\lim_{x\to-\infty}\frac{\sqrt{x^2+4}}x$

Find the hole, vertical asymptote, and horizontal asymptote of $\dfrac{x^2-1}{x^2-3x+2}$.

Find $k$ so $\begin{cases}kx+1,&x<2,\\x^2-1,&x\ge2\end{cases}$ is continuous at $2$.

Use IVT to show $x^3+x-3=0$ has a root in $(1,2)$.

Give a valid $\delta$ in terms of $\varepsilon$ for $\lim_{x\to3}(2x+1)=7$.