함수의 극한과 연속 전체 문제

In the following exercises, assume that $\lim_{x\to 6}f(x)=4,\lim_{x\to 6}g(x)=9,$ and $\lim_{x\to 6}h(x)=6.$ Use these three facts and the limit laws to evaluate each limit.

$\lim_{x\to 6}[(x+1)\cdot f(x)]$

In the following exercises, assume that $\lim_{x\to 6}f(x)=4,\lim_{x\to 6}g(x)=9,$ and $\lim_{x\to 6}h(x)=6.$ Use these three facts and the limit laws to evaluate each limit.

$\lim_{x\to 6}(f(x)\cdot g(x)-h(x))$

[T] In the following exercises, use a calculator to draw the graph of each piecewise-defined function and study the graph to evaluate the given limits.

$f(x)=\left\{\begin{array}{ll}{x}^{2}, & x\le 3 \\ x+4, & x>3\end{array}\right.$

a. $\lim_{x\to {3}^{-}}f(x)$ b. $\lim_{x\to {3}^{+}}f(x)$

[T] In the following exercises, use a calculator to draw the graph of each piecewise-defined function and study the graph to evaluate the given limits.

$g(x)=\left\{\begin{array}{ll}{x}^{3}-1, & x\le 0 \\ 1, & x>0\end{array}\right.$

a. $\lim_{x\to {0}^{-}}g(x)$ b. $\lim_{x\to {0}^{+}}g(x)$

In the following exercises, use the following graphs and the limit laws to evaluate each limit.

$\lim_{x\to {-3}^{+}}(f(x)+g(x))$

In the following exercises, use the following graphs and the limit laws to evaluate each limit.

$\lim_{x\to {-3}^{-}}(f(x)-3g(x))$

In the following exercises, use the following graphs and the limit laws to evaluate each limit.

$\lim_{x\to 0}\frac{f(x)g(x)}{3}$

In the following exercises, use the following graphs and the limit laws to evaluate each limit.

$\lim_{x\to -5}\frac{2+g(x)}{f(x)}$

In the following exercises, use the following graphs and the limit laws to evaluate each limit.

$\lim_{x\to 1}{(f(x))}^{2}$

In the following exercises, use the following graphs and the limit laws to evaluate each limit.

$\lim_{x\to 1}\sqrt[3]{f(x)-g(x)}$

In the following exercises, use the following graphs and the limit laws to evaluate each limit.

$\lim_{x\to -7}(x\cdot g(x))$

In the following exercises, use the following graphs and the limit laws to evaluate each limit.

$\lim_{x\to -9}[x\cdot f(x)+2\cdot g(x)]$

For the following problems, evaluate the limit using the squeeze theorem. Use a calculator to graph the functions $f(x),g(x),$ and $h(x)$ when possible.

[T] True or False? If $2x-1\le g(x)\le {x}^{2}-2x+3,$ then $\lim_{x\to 2}g(x)=0.$

For the following problems, evaluate the limit using the squeeze theorem. Use a calculator to graph the functions $f(x),g(x),$ and $h(x)$ when possible.

[T] $\lim_{\theta \to 0}{\theta }^{2}\cos \left(\frac{1}{\theta }\right)$

For the following problems, evaluate the limit using the squeeze theorem. Use a calculator to graph the functions $f(x),g(x),$ and $h(x)$ when possible.

$\lim_{x\to 0}f(x),$ where $f(x)=\left\{\begin{array}{ll}0, & x\,\text{rational} \\ {x}^{2}, & x\,\text{irrational}\end{array}\right.$

[T] The density of an object is given by its mass divided by its volume: $\rho =m/V.$

a. Use a calculator to plot the volume as a function of density $(V=m/\rho ),$ assuming you are examining something of mass 8 kg ($m=8).$ b. Evaluate $\lim_{\rho \to {0}^{+}}V(\rho )$ and explain the physical meaning.

For the following exercises, determine the point(s), if any, at which each function is discontinuous. Classify any discontinuity as jump, removable, infinite, or other.

$f(x)=\frac{1}{\sqrt{x}}$

For the following exercises, determine the point(s), if any, at which each function is discontinuous. Classify any discontinuity as jump, removable, infinite, or other.

$f(x)=\frac{2}{{x}^{2}+1}$

For the following exercises, determine the point(s), if any, at which each function is discontinuous. Classify any discontinuity as jump, removable, infinite, or other.

$f(x)=\frac{x}{{x}^{2}-x}$

For the following exercises, determine the point(s), if any, at which each function is discontinuous. Classify any discontinuity as jump, removable, infinite, or other.

$g(t)={t}^{-1}+1$