[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]

If [math]\lim_{x\goesto a} f(x) = \lim_{x\goesto a} g(x) = \pm \infty[/math], it is not immediately apparent whether or not [math]\lim_{x\goesto a} (f(x)-g(x))[/math] exists. Such limits are commonly called indeterminate forms of the type [math]\infty - \infty[/math]. The usual method of evaluation is to express the difference [math]f(x) - g(x)[/math] as a quotient and then to try to find its limit. For example, we write

[[math]] \frac{e^x}{e^x-1} - \frac1x = \frac{xe^x - (e^x-1)}{x(e^x - 1)} , [[/math]]

and, as [math]x[/math] approaches zero, the limit of the right side can be obtained by two applications of L'H\^opital's Rule. Evaluate

• [math]\lim_{x\goesto0} \left(\frac{e^x}{e^x-1} - \frac1x\right)[/math]
• [math]\lim_{x\goesto0} \left[\frac{(x^2+8)^{\frac13}} {2x^2} - \frac1{x^2}\right][/math]
• [math]\lim_{x\goesto0} \left(\frac{x^2+3x+5} {\sin x} - \frac5x\right)[/math]
• [math]\lim_{t\goesto0} \left(\cot t - \frac{1-2t}t\right)[/math]
• [math]\lim_{x\goesto0+} \left(\frac1x + \ln x\right)[/math]
• [math]\lim_{x\goesto\frac{\pi}2} (\sec x - \tan x)[/math].