5.3: Límites al Infinito y Límites Infinitos
- Page ID
- 108785
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Dejar\(D \subset \mathbb{R}, f: D \rightarrow \mathbb{R},\) y suponer\(a\) es un punto límite de\(D\). Decimos que\(f\) diverge a\(+\infty\) como\(x\) enfoques\(a\), denotado
\[\lim _{x \rightarrow a} f(x)=+\infty ,\]
si por cada número real\(M\) existe\(\delta>0\) tal que
\[f(x)>M \text { whenever } x \neq a \text { and } x \in(a-\delta, a+\delta) \cap D.\]
De igual manera, decimos que eso\(f\) diverge a\(-\infty\) como\(x\) enfoques\(a,\) denotados
\[\lim _{x \rightarrow a} f(x)=-\infty ,\]
si por cada número real\(M\) existe\(\delta>0\) tal que
\[f(x)<M \text { whenever } x \neq a \text { and } x \in(a-\delta, a+\delta) \cap D.\]
Proporcionar definiciones para
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\(\lim _{x \rightarrow a^{-}} f(x)=-\infty\).
Modele sus definiciones sobre las definiciones anteriores.
Demuestre eso\(\lim _{x \rightarrow 4^{+}} \frac{7}{4-x}=-\infty\) y\(\lim _{x \rightarrow 4^{-}} \frac{7}{4-x}=+\infty\).
Supongamos que\(D \subset \mathbb{R}\) no tiene un límite superior\(f: D \rightarrow \mathbb{R}\),, y\(L \in \mathbb{R} .\) Nosotros decimos que el límite de\(f\) como\(+\infty\) se\(x\) acerca se\(L,\) denota
\[\lim _{x \rightarrow+\infty} f(x)=L,\]
si por cada\(\epsilon>0\) existe un número real\(M\) tal que
\[|f(x)-L|<\epsilon \text { whenever } x \in(M,+\infty) \cap D.\]
Supongamos que\(D \subset \mathbb{R}\) no tiene un límite inferior\(f: D \rightarrow \mathbb{R}\),, y\(L \in \mathbb{R} .\) Nosotros decimos que el límite de\(f\) como\(-\infty\) se\(x\) acerca se\(L,\) denota
\[\lim _{x \rightarrow-\infty} f(x)=L,\]
si por cada\(\epsilon>0\) existe un número real\(M\) tal que
\[|f(x)-L|<\epsilon \text { whenever } x \in(-\infty, M) \cap D.\]
Verifica eso\(\lim _{x \rightarrow+\infty} \frac{x+1}{x+2}=1\).
Proporcionar definiciones para
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\(\lim _{x \rightarrow-\infty} f(x)=-\infty\).
Modele sus definiciones sobre las definiciones anteriores.
Supongamos
\[f(x)=a x^{3}+b x^{2}+c x+d,\]
donde\(a, b, c, d \in \mathbb{R}\) y\(a>0 .\) Demostrar que
\[\lim _{x \rightarrow+\infty} f(x)=+\infty \text { and } \lim _{x \rightarrow-\infty} f(x)=-\infty .\]