⧼exchistory⧽
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BBy Bot
Nov 03'24
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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].
BBy Bot
Nov 03'24
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Evaluate each of the limits in Problem Exercise using an appropriate for of L'H\^opital's Rule, if it is applicable.

BBy Bot
Nov 03'24
[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.

  • [math]\lim_{x \goesto 4} \frac{x-4}{x^n-4^n}[/math], [math]n[/math] is a positive integer
  • [math]\lim_{x \goesto 1+} \frac{x^\frac32 - 1}{\sqrt{x^3-1}}[/math]
  • [math]\lim_{x \goesto 2+} \frac{x^2 - 4x + 2}{\sqrt{x^2 - 4}}[/math]
  • [math]\lim_{x \goesto 1} \frac{x^\frac12-x^\frac13}{x-1}[/math]
  • [math]\lim_{x \goesto 1} \frac{x^3-x^2-x+1}{2x^3-3x^2+1}[/math]
  • [math]\lim_{t \goesto 0} \frac{3t^2}{3(1+t)^\frac13-t-3}[/math].
BBy Bot
Nov 03'24
[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]

Compute

  • [math]\lim_{x \goesto \infty} \frac{\sqrt{x+1}}{x+2}[/math]
  • [math]\lim_{x \goesto \infty} \frac{(x^2+1)^\frac13}{2x^2-3}[/math]
  • [math]\lim_{x \goesto \infty} \frac{x^\frac13 + 2}{x^\frac12 - 2}[/math]
  • [math]\lim_{x \goesto \infty} \frac{x^\frac13+ 2x+1}{x^\frac12+3x-2}[/math].
BBy Bot
Nov 03'24
[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]

Compute each of the following limits directly using the [math]\frac*\infty[/math] form of L'H\^opital's Rule. Verify the result by writing the quotient in a different form and using either the [math]\frac00[/math] form of the rule or some other method.

  • [math]\lim_{x \goesto 0+} \frac{\sqrt{\frac1x + 1}}{\frac1x+2}[/math]
  • [math]\lim_{x \goesto 0} \frac {\frac1{x^2}+5} {\left( \frac1{x^2}-1 \right)^\frac13}[/math].
BBy Bot
Nov 03'24
[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]

Suppose that [math]F[/math] is a function differentiable on the open interval [math](0, \infty)[/math] and such that [math]F^\prime(x) = \frac1x[/math], for every [math]x \gt 0[/math]. Show that

  • [math]\lim_{x \goesto \infty} \frac{F(x)}x = 0[/math]
  • [math]\lim_{x \goesto \infty} \frac{F(x)}{x^2} = 0[/math]
  • [math]\lim_{x \goesto \infty} \frac{F(x)}{x^n} = 0[/math], for every positive integer [math]n[/math].