[math] \newcommand{\ex}[1]{\item } \newcommand{\sx}{\item} \newcommand{\x}{\sx} \newcommand{\sxlab}[1]{} \newcommand{\xlab}{\sxlab} \newcommand{\prov}[1] {\quad #1} \newcommand{\provx}[1] {\quad \mbox{#1}} \newcommand{\intext}[1]{\quad \mbox{#1} \quad} \newcommand{\R}{\mathrm{\bf R}} \newcommand{\Q}{\mathrm{\bf Q}} \newcommand{\Z}{\mathrm{\bf Z}} \newcommand{\C}{\mathrm{\bf C}} \newcommand{\dt}{\textbf} \newcommand{\goesto}{\rightarrow} \newcommand{\ddxof}[1]{\frac{d #1}{d x}} \newcommand{\ddx}{\frac{d}{dx}} \newcommand{\ddt}{\frac{d}{dt}} \newcommand{\dydx}{\ddxof y} \newcommand{\nxder}[3]{\frac{d^{#1}{#2}}{d{#3}^{#1}}} \newcommand{\deriv}[2]{\frac{d^{#1}{#2}}{dx^{#1}}} \newcommand{\dist}{\mathrm{distance}} \newcommand{\arccot}{\mathrm{arccot\:}} \newcommand{\arccsc}{\mathrm{arccsc\:}} \newcommand{\arcsec}{\mathrm{arcsec\:}} \newcommand{\arctanh}{\mathrm{arctanh\:}} \newcommand{\arcsinh}{\mathrm{arcsinh\:}} \newcommand{\arccosh}{\mathrm{arccosh\:}} \newcommand{\sech}{\mathrm{sech\:}} \newcommand{\csch}{\mathrm{csch\:}} \newcommand{\conj}[1]{\overline{#1}} \newcommand{\mathds}{\mathbb} [/math]

Evaluate each of the following limits without using L'H\^opital's Rule.

• [math]\lim_{x \goesto 3} \frac{x^2 - 9}{x^2 - 5x + 6}[/math]
• [math]\lim_{x \goesto -2} \frac{x^3 + 8}{x^5+32}[/math]
• [math]\lim_{x \goesto 2} \frac{x^3 - 6x + 4}{x^2 + 4}[/math]
• [math]\lim_{x \goesto \infty} \frac{2x^2 + x - 1}{3x^2 - 2x + 1}[/math]
• [math]\lim_{x \goesto 1} \frac{x^2 - 1}{x^3 - 1}[/math]
• [math]\lim_{x \goesto 3} \frac{\sqrt x - \sqrt3}{x-3}[/math]
• [math]\lim_{t \goesto 2} \frac{t^2 + t + 6}{t^3 - 2t + 4}[/math]
• [math]\lim_{t \goesto 0} \frac t{\sqrt{1+t}-1}[/math].