Cumulative Limits and Continuity Practice

Explain the difference between $f(a)$ and $\lim_{x\to a}f(x)$.

State the one-sided criterion for a two-sided limit.

List four reasons a limit may fail to exist.

State the three continuity conditions at $a$.

Explain why changing one value can repair a hole but not a jump.

Explain why infinity is not treated as an ordinary real-number limit.

State the Squeeze Theorem.

State the fundamental sine limit and its unit requirement.

State the Intermediate Value Theorem, including every hypothesis.

Explain what IVT does not guarantee.

$\displaystyle\lim_{x\to2}(3x^2-x+4)$

$\displaystyle\lim_{x\to-1}\frac{x^2+2}{x+4}$

$\displaystyle\lim_{x\to4}\frac{x^2-16}{x-4}$

$\displaystyle\lim_{x\to-2}\frac{x^3+8}{x+2}$

$\displaystyle\lim_{x\to1}\frac{x^4-1}{x^2-1}$

$\displaystyle\lim_{x\to0}\frac{\sqrt{x+25}-5}{x}$

$\displaystyle\lim_{x\to9}\frac{x-9}{\sqrt{x}-3}$

$\displaystyle\lim_{x\to0}\frac{\frac1{x+2}-\frac12}{x}$

$\displaystyle\lim_{x\to3}\frac{|x-3|}{x-3}$

$\displaystyle\lim_{x\to2}\frac{|x-2|}{\sqrt{|x-2|}}$

$\displaystyle\lim_{x\to0}x^3\sin(1/x)$

$\displaystyle\lim_{x\to0}x^2\cos(4/x)$

$\displaystyle\lim_{x\to0}\frac{\sin(6x)}x$

$\displaystyle\lim_{x\to0}\frac{\sin(2x)}{\sin(9x)}$

$\displaystyle\lim_{x\to0}\frac{\tan(5x)}{3x}$

$\displaystyle\lim_{x\to0}\frac{1-\cos(4x)}{x^2}$

$\displaystyle\lim_{x\to0}\frac{\sin(3x)\sin(7x)}{x^2}$

$\displaystyle\lim_{x\to0}\frac{1-\cos x}{x\sin x}$

$\displaystyle\lim_{x\to1^-}\frac{x+2}{x-1}$

$\displaystyle\lim_{x\to1^+}\frac{x+2}{x-1}$

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

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

$\displaystyle\lim_{x\to\infty}\frac{4x^2-x}{2x^2+7}$

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

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

Find the slant asymptote of $\dfrac{x^2+2}{x-1}$.

$\displaystyle\lim_{x\to\infty}\frac{\sqrt{4x^2+1}}x$

$\displaystyle\lim_{x\to-\infty}\frac{\sqrt{4x^2+1}}x$

$\displaystyle\lim_{x\to\infty}(\sqrt{x^2+12x}-x)$

$\displaystyle\lim_{x\to-\infty}(\sqrt{x^2+8x}+x)$

Find intervals of continuity of $\sqrt{x+1}/(x-2)$.

Classify all discontinuities of $(x^2-4)/(x^2-x-2)$.

Find $c$ so $\begin{cases}(x^2-9)/(x-3),&x\ne3,\\c,&x=3\end{cases}$ is continuous.

Find $k$ so $\begin{cases}kx+2,&x<1,\\x^2+4,&x\ge1\end{cases}$ is continuous at $1$.

Determine whether any $a$ makes $\begin{cases}ax+1,&x<0,\\ax+4,&x\ge0\end{cases}$ continuous at $0$.

Show that $x^3-x-1=0$ has a root in $(1,2)$.

Explain why $1/x$ cannot use IVT on $[-1,1]$.

Use two bisection steps to narrow a root of $x^3-x-1$ from $(1,2)$.

Find a $\delta$ in terms of $\varepsilon$ proving $\lim_{x\to2}(4x-1)=7$.

Prove $\lim_{x\to1}x^2=1$ using a local bound.

Explain why $\delta=\varepsilon$ does not automatically work for every function.

Translate $|x-5|<0.01$ and $|f(x)-3|<0.02$ into interval language.