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Mathematics LibreTexts

5.3: Limits to Infinity and Infinite Limits

  • Page ID
    22666
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    Definition

    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

    \[f(x)<M \text { whenever } x \neq a \text { and } x \in(a-\delta, a+\delta) \cap D.\]

    Exercise \(\PageIndex{1}\)

    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.

    Exercise \(\PageIndex{2}\)

    Show that \(\lim _{x \rightarrow 4^{+}} \frac{7}{4-x}=-\infty\) and \(\lim _{x \rightarrow 4^{-}} \frac{7}{4-x}=+\infty\).

    Definition

    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

    \[|f(x)-L|<\epsilon \text { whenever } x \in(M,+\infty) \cap D.\]

    Definition

    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

    \[|f(x)-L|<\epsilon \text { whenever } x \in(-\infty, M) \cap D.\]

    Exercise \(\PageIndex{3}\)

    Verify that \(\lim _{x \rightarrow+\infty} \frac{x+1}{x+2}=1\).

    Exercise \(\PageIndex{4}\)

    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.

    Exercise \(\PageIndex{5}\)

    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

    \[\lim _{x \rightarrow+\infty} f(x)=+\infty \text { and } \lim _{x \rightarrow-\infty} f(x)=-\infty .\]

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