5.3: Limits to Infinity and Infinite Limits
- Page ID
- 22666
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Let \(D \subset \mathbb{R}, f: D \rightarrow \mathbb{R},\) and suppose \(a\) is a limit point of \(D\). We say that \(f\) diverges to \(+\infty\) as \(x\) approaches \(a\), denoted
\[\lim _{x \rightarrow a} f(x)=+\infty ,\]
if for every real number \(M\) there exists a \(\delta>0\) such that
\[f(x)>M \text { whenever } x \neq a \text { and } x \in(a-\delta, a+\delta) \cap D.\]
Similarly, we say that that \(f\) diverges to \(-\infty\) as \(x\) approaches \(a,\) denoted
\[\lim _{x \rightarrow a} f(x)=-\infty ,\]
if for every real number \(M\) there exists a \(\delta>0\) such that
\[f(x)<M \text { whenever } x \neq a \text { and } x \in(a-\delta, a+\delta) \cap D.\]
Provide definitions for
a. \(\lim _{x \rightarrow a^{+}} f(x)=+\infty\),
b. \(\lim _{x \rightarrow a^{-}} f(x)=+\infty\),
c. \(\lim _{x \rightarrow a^{+}} f(x)=-\infty\),
d. \(\lim _{x \rightarrow a^{-}} f(x)=-\infty\).
Model your definitions on the preceding definitions.
Show that \(\lim _{x \rightarrow 4^{+}} \frac{7}{4-x}=-\infty\) and \(\lim _{x \rightarrow 4^{-}} \frac{7}{4-x}=+\infty\).
Suppose \(D \subset \mathbb{R}\) does not have an upper bound, \(f: D \rightarrow \mathbb{R}\), and \(L \in \mathbb{R} .\) We say that the limit of \(f\) as \(x\) approaches \(+\infty\) is \(L,\) denoted
\[\lim _{x \rightarrow+\infty} f(x)=L,\]
if for every \(\epsilon>0\) there exists a real number \(M\) such that
\[|f(x)-L|<\epsilon \text { whenever } x \in(M,+\infty) \cap D.\]
Suppose \(D \subset \mathbb{R}\) does not have an lower bound, \(f: D \rightarrow \mathbb{R}\), and \(L \in \mathbb{R} .\) We say that the limit of \(f\) as \(x\) approaches \(-\infty\) is \(L,\) denoted
\[\lim _{x \rightarrow-\infty} f(x)=L,\]
if for every \(\epsilon>0\) there exists a real number \(M\) such that
\[|f(x)-L|<\epsilon \text { whenever } x \in(-\infty, M) \cap D.\]
Verify that \(\lim _{x \rightarrow+\infty} \frac{x+1}{x+2}=1\).
Provide definitions for
a. \(\lim _{x \rightarrow+\infty} f(x)=+\infty\),
b. \(\lim _{x \rightarrow+\infty} f(x)=-\infty\),
c. \(\lim _{x \rightarrow-\infty} f(x)=+\infty\),
d. \(\lim _{x \rightarrow-\infty} f(x)=-\infty\).
Model your definitions on the preceding definitions.
Suppose
\[f(x)=a x^{3}+b x^{2}+c x+d,\]
where \(a, b, c, d \in \mathbb{R}\) and \(a>0 .\) Show that
\[\lim _{x \rightarrow+\infty} f(x)=+\infty \text { and } \lim _{x \rightarrow-\infty} f(x)=-\infty .\]