$54$

Chapter $2$ Limits and Continuity

$31.\underset{x1}{lim}\frac{x^{-1}-1}{x-1}$

$32.\underset{x0}{lim}\frac{\frac{1}{x-1}+\frac{1}{x+1}}{x}$

$33.\underset{u1}{lim}\frac{u^4-1}{u^3-1}$

$34.\underset{v2}{lim}\frac{v^3-8}{v^4-16}$

$35.\underset{x9}{lim}\frac{\sqrt[\phantom{2}]{x}-3}{x-9}$

$36.\underset{x4}{lim}\frac{4x-x^2}{2-\sqrt[\phantom{2}]{x}}$

$37.\underset{x1}{lim}\frac{x-1}{\sqrt[\phantom{2}]{x+3}-2}$

$38.\underset{x-1}{lim}\frac{\sqrt[\phantom{2}]{x^2+8}-3}{x+1}$

$39.\underset{x2}{lim}\frac{\sqrt[\phantom{2}]{x^2+12}-4}{x-2}$

$40.\underset{x-2}{lim}\frac{x+2}{\sqrt[\phantom{2}]{x^2+5}-3}$

$41.\underset{x-3}{lim}\frac{2-\sqrt[\phantom{2}]{x^2-5}}{x+3}$

$42.\underset{x4}{lim}\frac{4-x}{5-\sqrt[\phantom{2}]{x^2+9}}$

Limits with trigonometric functions Find the limits in Exercises $43-50.$

$43.\underset{x0}{lim}(2sinx-1)$

$44.\underset{x0}{lim}si{n}^{2}x$

$45.\underset{x0}{lim}secx$

$46.\underset{x0}{lim}tanx$

$47.\underset{x0}{lim}\frac{1+x+sinx}{3cosx}$

$48.\underset{x0}{lim}(x^2-1)(2-cosx)$

$49.\underset{x-}{lim}\sqrt[\phantom{2}]{x+4}cos(x+)$

$50.\underset{x0}{lim}\sqrt[\phantom{2}]{7+se{c}^{2}x}$

Using Limit Rules

$51.$ Suppose $\underset{x0}{lim}f(x)=1$ and $\underset{x0}{lim}g(x)=-5.$ Name the rules in Theorem $1$ that are used to accomplish steps $($a$),($b$),$ and $($c$)$ of the following calculation.

$\underset{x0}{lim}\frac{2f(x)-g(x)}{(f(x)+7{)}^2}=\frac{\underset{x0}{lim}(2f(x)-g(x))}{\underset{x0}{lim}(f(x)+7{)}^2}$

$($We assume the denominator is nonzero.$)$

$=\frac{\underset{x0}{lim}2f(x)-\underset{x0}{lim}g(x)}{(\underset{x0}{lim}(f(x)+7){)}^2}$

$=\frac{2\underset{x0}{lim}f(x)-\underset{x0}{lim}g(x)}{(\underset{x0}{lim}f(x)+\underset{x0}{lim}7{)}^2}$

$=\frac{(2)(1)-(-5)}{(1+7{)}^2}=\frac{7}{64}$