2.3: The Limit Laws

Last updated

Nov 30, 2023
Save as PDF
- 2.2: The Limit of a Function
- 2.4: The Precise Definition of a Limit

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\id}{\mathrm{id}}$ $\newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$ $\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$ $\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\id}{\mathrm{id}}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\kernel}{\mathrm{null}\,}$

$\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$

$\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$

$\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$ $\newcommand{\AA}{\unicode[.8,0]{x212B}}$

$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vectorC}[1]{\textbf{#1}}$

$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$

$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$

$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\avec}{\mathbf a}$

$\newcommand{\bvec}{\mathbf b}$

$\newcommand{\cvec}{\mathbf c}$

$\newcommand{\dvec}{\mathbf d}$

$\newcommand{\dtil}{\widetilde{\mathbf d}}$

$\newcommand{\evec}{\mathbf e}$

$\newcommand{\fvec}{\mathbf f}$

$\newcommand{\nvec}{\mathbf n}$

$\newcommand{\pvec}{\mathbf p}$

$\newcommand{\qvec}{\mathbf q}$

$\newcommand{\svec}{\mathbf s}$

$\newcommand{\tvec}{\mathbf t}$

$\newcommand{\uvec}{\mathbf u}$

$\newcommand{\vvec}{\mathbf v}$

$\newcommand{\wvec}{\mathbf w}$

$\newcommand{\xvec}{\mathbf x}$

$\newcommand{\yvec}{\mathbf y}$

$\newcommand{\zvec}{\mathbf z}$

$\newcommand{\rvec}{\mathbf r}$

$\newcommand{\mvec}{\mathbf m}$

$\newcommand{\zerovec}{\mathbf 0}$

$\newcommand{\onevec}{\mathbf 1}$

$\newcommand{\real}{\mathbb R}$

$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$

$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$

$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$

$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$

$\newcommand{\bcal}{\cal B}$

$\newcommand{\ccal}{\cal C}$

$\newcommand{\scal}{\cal S}$

$\newcommand{\wcal}{\cal W}$

$\newcommand{\ecal}{\cal E}$

$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$

$\newcommand{\gray}[1]{\color{gray}{#1}}$

$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$

$\newcommand{\rank}{\operatorname{rank}}$

$\newcommand{\row}{\text{Row}}$

$\newcommand{\col}{\text{Col}}$

$\renewcommand{\row}{\text{Row}}$

$\newcommand{\nul}{\text{Nul}}$

$\newcommand{\var}{\text{Var}}$

$\newcommand{\corr}{\text{corr}}$

$\newcommand{\len}[1]{\left|#1\right|}$

$\newcommand{\bbar}{\overline{\bvec}}$

$\newcommand{\bhat}{\widehat{\bvec}}$

$\newcommand{\bperp}{\bvec^\perp}$

$\newcommand{\xhat}{\widehat{\xvec}}$

$\newcommand{\vhat}{\widehat{\vvec}}$

$\newcommand{\uhat}{\widehat{\uvec}}$

$\newcommand{\what}{\widehat{\wvec}}$

$\newcommand{\Sighat}{\widehat{\Sigma}}$

$\newcommand{\lt}{<}$

$\newcommand{\gt}{>}$

$\newcommand{\amp}{&}$

$\definecolor{fillinmathshade}{gray}{0.9}$

Evaluate $\displaystyle \lim_{x \to 2} 2x^2$ , if possible.
Evaluate $\displaystyle \lim_{x \to -3} 7$ , if possible.
Evaluate $\displaystyle \lim_{x \to 5} (3x - 12)$ , if possible.
Evaluate $\displaystyle \lim_{x \to 2} \dfrac{x+7}{x-3}$ , if possible.
Evaluate $\displaystyle \lim_{x \to 1} \dfrac{\ln x}{\cos(\pi x)}$ , if possible.
Evaluate $\displaystyle \lim_{x \to -1} \sqrt{x^2 - 7x - 2}$ , if possible.
Evaluate $\displaystyle \lim_{x \to 0} \dfrac{x}{x}$ , if possible.
Evaluate $\displaystyle \lim_{x \to -3} \dfrac{x^2 - 9}{x + 3}$ , if possible.
Evaluate $\displaystyle \lim_{x \to 4} \dfrac{x^2 - 6x + 8}{x^2 + x - 20}$ , if possible.
Evaluate $\displaystyle \lim_{h \to 0} \dfrac{(x + h)^2 - x^2}{h}$ , if possible.
Evaluate $\displaystyle \lim_{t \to 4} \dfrac{\sqrt{t} - 2}{t - 4}$ , if possible.
Evaluate $\displaystyle \lim_{\theta \to \pi} \dfrac{\sin \theta}{\tan \theta}$ , if possible.
Evaluate $\displaystyle \lim_{h \to 0} \dfrac{\dfrac{1}{x+h} - \dfrac{1}{x}}{h}$ , if possible.
Evaluate $\displaystyle \lim_{x \to 5} \dfrac{x - 5}{3 - \sqrt{x + 4}}$ , if possible.
Evaluate $\displaystyle \lim_{x \to 0} \sin\left(\dfrac{1}{x}\right)$ , if possible.
Evaluate $\displaystyle \lim_{x \to -2^-} \dfrac{x^2-1}{x^2-x-6}$ , if possible.
Evaluate $\displaystyle \lim_{x \to -2^+} \dfrac{x^2-1}{x^2-x-6}$ , if possible.
Evaluate $\displaystyle \lim_{x \to -2} \dfrac{x^2-1}{x^2-x-6}$ , if possible.
Evaluate $\displaystyle \lim_{x \to 3^-} \dfrac{x^2-1}{x^2-x-6}$ , if possible.
Evaluate $\displaystyle \lim_{x \to 3^+} \dfrac{x^2-1}{x^2-x-6}$ , if possible.
Evaluate $\displaystyle \lim_{x \to 3} \dfrac{x^2-1}{x^2-x-6}$ , if possible.
Evaluate $\displaystyle \lim_{x \to 1^-} \dfrac{1}{x^2 - 2x + 1}$ , if possible.
Evaluate $\displaystyle \lim_{x \to 1^+} \dfrac{1}{x^2 - 2x + 1}$ , if possible.
Evaluate $\displaystyle \lim_{x \to 1} \dfrac{1}{x^2 - 2x + 1}$ , if possible.
Evaluate $\displaystyle \lim_{x \to 0^+} \sqrt{x}$ , if possible.
Evaluate $\displaystyle \lim_{x \to 0^+} \ln x$ , if possible.
Consider the function
$f(x) = \left\{\begin{align*} & x^2, & x < 1, \\ & x, & x \ge 1. \end{align*}\right.$
Find $\displaystyle\lim_{x \to 1^-} f(x)$ , $\displaystyle\lim_{x \to 1^+} f(x)$ , and $\displaystyle\lim_{x \to 1} f(x)$ , if possible.
Consider the function
$g(x) = \left\{\begin{align*} & \cos x, & x \le 0, \\ & \sin x, & x > 0. \end{align*}\right.$
Find $\displaystyle\lim_{x \to 0^-} g(x)$ , $\displaystyle\lim_{x \to 0^+} g(x)$ , and $\displaystyle\lim_{x \to 0} g(x)$ , if possible.
Consider the function
$h(x) = \left\{\begin{align*} & \dfrac{1}{x+1}, & x < -1, \\ & \dfrac{1}{x}, & -1 \le x < 0. \end{align*}\right.$
Find $\displaystyle\lim_{x \to -1^-} h(x)$ , $\displaystyle\lim_{x \to -1^+} h(x)$ , and $\displaystyle\lim_{x \to -1} h(x)$ , if possible.
Consider the function
$f(x) = \left\{\begin{align*} & |1 - x^2|, & x < 0, \\ & \sqrt{2}, & x = 0, \\ & e^x, & x > 0. \end{align*}\right.$
Find $\displaystyle\lim_{x \to 0^-} f(x)$ , $\displaystyle\lim_{x \to 0^+} f(x)$ , and $\displaystyle\lim_{x \to 0} f(x)$ , if possible.

Search

Text Color

Text Size

Margin Size

Font Type

Support Center

How can we help?