극한의 성질과 계산

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.

A function has to be continuous at $x=a$ if the $\lim_{x\to a}f(x)$ exists.

You can use the quotient rule to evaluate $\lim_{x\to 0}\frac{\sin x}{x}.$

If there is a vertical asymptote at $x=a$ for the function $f(x),$ then f is undefined at the point $x=a.$

If $\lim_{x\to a}f(x)$ does not exist, then f is undefined at the point $x=a.$

Using the graph, find each limit or explain why the limit does not exist.

a. $\lim_{x\to -1}f(x)$ b. $\lim_{x\to 1}f(x)$ c. $\lim_{x\to {0}^{+}}f(x)$ d. $\lim_{x\to 2}f(x)$

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.

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

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.

$\lim_{x\to 0}3{x}^{2}-2x+4$

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.

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

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.

$\lim_{x\to \pi /2}\frac{\cot x}{\cos x}$

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.

$\lim_{x\to -5}\frac{{x}^{2}+25}{x+5}$

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.

$\lim_{x\to 2}\frac{3{x}^{2}-2x-8}{{x}^{2}-4}$

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.

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

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.

$\lim_{x\to 1}\frac{{x}^{2}-1}{\sqrt{x}-1}$

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.

$\lim_{x\to 4}\frac{4-x}{\sqrt{x}-2}$

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.

$\lim_{x\to 4}\frac{1}{\sqrt{x}-2}$