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.

$\lim_{x\to 6}2f(x)g(x)$

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.

$\lim_{x\to 6}\frac{g(x)-1}{f(x)}$

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.

$\lim_{x\to 6}\left(f(x)+\frac{1}{3}g(x)\right)$

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.

$\lim_{x\to 6}\frac{{(h(x))}^{3}}{2}$

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.

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

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.

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

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.

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

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] 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.

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

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.

$f(x)=\frac{5}{{e}^{x}-2}$

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.

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

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.

$H(x)=\tan 2x$

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.

$f(t)=\frac{t+3}{{t}^{2}+5t+6}$

For the following exercises, decide if the function continuous at the given point. If it is discontinuous, what type of discontinuity is it?

$h(\theta )=\frac{\sin \theta -\cos \theta }{\tan \theta }$ at $\theta =\pi$

For the following exercises, decide if the function continuous at the given point. If it is discontinuous, what type of discontinuity is it?

$g(u)=\left\{\begin{array}{ll}\frac{6{u}^{2}+u-2}{2u-1} & \text{if}\,u\ne \frac{1}{2} \\ \frac{7}{2} & \text{if}\,u=\frac{1}{2}\end{array}\right.,$ at $u=\frac{1}{2}$

For the following exercises, decide if the function continuous at the given point. If it is discontinuous, what type of discontinuity is it?

$f(y)=\frac{\sin (\pi y)}{\tan (\pi y)},$ at $y=1$