91Ó°ÊÓ

In Problems 53-56, use the properties of limits and the facts that

$$\begin{gathered} \underset{x→0}{\lim }\frac{\sin x}{x}=1 \\ \underset{x→0}{\lim }\frac{\cos x-1}{x}=1 \\ \underset{x→0}{\lim }\sin x=0 \\ \underset{x→0}{\lim }\cos x=1 \end{gathered}$$

$\underset{x→0}{\lim }\frac{3\sin x+\cos x-1}{4x}$

Use a graphing utility to find the indicated limit rounded to two decimal places.

$\underset{x→2}{\lim }\frac{x^3-2x^2+4x-8}{x^2+x-6}$

Determine whether f is continuous at c.

$f\left(x\right)=\left\{\begin{array}{l}\frac{x^3+3x}{x^2−3x}â€\dots â\P Ä\dots â\P Ä\dots â\P Ä\dots \text{if}x≠0\\ −1â€\dots â\P Ä\dots â\P Ä\dots â\P Ä\dots \text{if}x=0â€\dots â\P Ä\dots â\P Ä\dots â\P Ä\dots \end{array}c=0\right.$

Determine whether f is continuous at c.

$f\left(x\right)=\left\{\begin{array}{l}\frac{x^2−6x}{x^2+6x}â€\dots â\P Ä\dots â\P Ä\dots â\P Ä\dots \text{if}x≠0\\ −1â€\dots â\P Ä\dots â\P Ä\dots â\P Ä\dots \text{if}x=0\end{array}c=0\right.$

In Problems 53-56, use the properties of limits and the facts that

$$\begin{gathered} \underset{x→0}{\lim }\frac{\sin x}{x}=1 \\ \underset{x→0}{\lim }\frac{\cos x-1}{x}=0 \\ \underset{x→0}{\lim }\sin x=0 \\ \underset{x→0}{\lim }\cos x=1 \end{gathered}$$

$\underset{x→0}{\lim }\frac{{\sin }^{2}x+\sin x(\cos x-1)}{x^2}$

Determine whether f is continuous at c.

$f\left(x\right)=\left\{\begin{array}{l}\frac{x^3−1}{x^2−1}â€\dots â\P Ä\dots â\P Ä\dots â\P Ä\dots \text{if}x<1\\ 2â€\dots â\P Ä\dots â\P Ä\dots â\P Ä\dots \text{if}x=1c=1\\ \frac{3}{x+1}â€\dots â\P Ä\dots â\P Ä\dots â\P Ä\dots \text{if}x>1\end{array}\right.$

Determine whether f is continuous at c.

$f\left(x\right)=\left\{\begin{array}{l}\frac{x^2−2x}{x−2}â€\dots â\P Ä\dots â\P Ä\dots â\P Ä\dots \text{if}x<2\\ 2â€\dots â\P Ä\dots â\P Ä\dots â\P Ä\dots \text{if}x=2\\ \frac{x−4}{x−1}â€\dots â\P Ä\dots â\P Ä\dots â\P Ä\dots \text{if}x>2\end{array}c=2\right.$

Determine whether f is continuous at c.

$f\left(x\right)=\left\{\begin{array}{l}2e^xâ€\dots â\P Ä\dots â\P Ä\dots â\P Ä\dots \text{if}x<0\\ 2â€\dots â\P Ä\dots â\P Ä\dots â\P Ä\dots \text{if}x=0\\ \frac{x^3+2x^2}{x^2}â€\dots â\P Ä\dots â\P Ä\dots â\P Ä\dots \text{if}x>0\end{array}c=0\right.$

In Problems 5 and 6, refer to the illustration. The interval $[1,3]$is partitioned into two sub intervals $[1,2]$and $[2,3]$.

Approximate the area A choosing u as the right endpoint of
each sub interval.

True or False The limit of a quotient equals the quotient of the limits.