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Wize University Calculus 1 Textbook > Limits

Infinite Limits

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Infinite Limits

Infinite Limits describe the behavior of functions whose values become arbitrarily large either positively or negatively. The answer to these limits is ±∞\pm\infty.

(Positive) Infinite Limits

Let f(x)f\left(x\right) be a function defined on both sides of aa except possibly at aa itself. If f(x)f(x) grows arbitrarily large as xx approaches aa then
lim⁡x→af(x)=∞ \displaystyle \boxed{\lim_{x \to a} f(x)=\infty}

(Negative) Infinite Limits

Let f(x)f(x) be a function defined on both sides of aa except possibly at aa itself. If f(x)f(x) grows arbitrary small as xxapproaches aa then
lim⁡x→af(x)=−∞ \displaystyle \boxed{\lim_{x \to a} f(x)=-\infty}


Wize Tip
When trying to evaluate infinite limits at aa ,you will obtain a 0 in the denominator. Your job is to figure out if the function is growing positively or negatives close to aa.


Vertical Asymptotes

A function f(x)f\left(x\right) has a vertical asymptote with equation x=ax=a if at least one of these statements is true:

lim⁡x→a−f(x)=±∞    or    lim⁡x→a+f(x)=±∞\displaystyle \boxed{\lim_{x \to a^-} f(x)= \pm \infty \;\; \text{or}\;\; \lim_{x \to a^+} f(x)= \pm \infty }


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Example: Infinite Limits

For the following graph, find:

i) lim⁡x→5+f(x)\displaystyle \lim_{x \to 5^+}f(x)

ii)lim⁡x→5−f(x)\displaystyle \lim_{x \to 5^-}f(x)
Note: The graph above has a vertical asymptote at x=5x=5.

i) lim⁡x→5+f(x)=∞\displaystyle \lim_{x \to 5^+}f(x)=\infin

ii) lim⁡x→5−f(x)=−∞\displaystyle \lim_{x \to 5^-}f(x)=-\infin
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Example: Infinite Limits

Find the following limit

lim⁡x→73+7x−63x−7\displaystyle \lim_{x \to \frac{7}{3}^+} \frac{7x-6}{3x-7}


When we try to evaluate the limit, the numerator is positive while the denominator becomes 0. Evaluating with numbers to the right of 7/3 will give positive values in the denominator.

=493−60+=+0+=+∞=\displaystyle\frac{\frac{49}{3}-6}{0^+}=\frac{+}{0^+}=+\infty

Extra Practice
Limits: Practice
Find the following limit:
lim⁡x→2−5(x+1)x−2\displaystyle\lim_{x\rightarrow 2}\frac{-5(x+1)}{x-2}
Asymptotes of Exponential
Determine if the function g(x)=e−x2+2x−3g\left(x\right)=e^{-x^2+2x-3} has any horizontal asymptotes.
Infinite Limits
Evaluate lim⁡x→∞arctan⁡(x−x3)\displaystyle \lim_{x\to\infty}\arctan\left(x-x^3\right)
Infinite Limits
Evaluate lim⁡x→∞arctan⁡(x−x3)\displaystyle \lim_{x\to\infty}\arctan\left(x-x^3\right)
Infinite Limits
Determine lim⁡x→π−ln⁡(sin⁡x)\displaystyle\lim_{x \rightarrow\pi^-}\ln(\sin x) .
Practice: Log Limits
Determine lim⁡x →0 ln⁡(−ln⁡(x2))\lim_{x\ \rightarrow0}\ \ln\left(-\ln\left(x^2\right)\right).
Limits at infinity: Indeterminate forms
Which of the following statements is true about the functionf(x)=7x72+x12+15x72−5x12f\left(x\right)=\frac{7x^{\frac{7}{2}}+x^{\frac{1}{2}}+1}{5x^{\frac{7}{2}}-5x^{\frac{1}{2}}}?
Limits at infinity: Indeterminate forms
Which of the following statements is true about the functionf(x)=7x72+x12+15x72−5x12f\left(x\right)=\frac{7x^{\frac{7}{2}}+x^{\frac{1}{2}}+1}{5x^{\frac{7}{2}}-5x^{\frac{1}{2}}}?
Practice: Log Limits
Determine lim⁡x →0 ln⁡(−ln⁡(x2))\lim_{x\ \rightarrow0}\ \ln\left(-\ln\left(x^2\right)\right).
Practice: Log Limits
Determine lim⁡x →0 ln⁡(−ln⁡(x2))\lim_{x\ \rightarrow0}\ \ln\left(-\ln\left(x^2\right)\right).
Infinite Limits: Evaluating limits
Evaluate lim⁡ x→2+x−32−x\lim\ _{x\rightarrow2^+}\frac{x-3}{2-x}
Limits at infinity: Indeterminate forms
Which of the following statements is true about the functionf(x)=7x72+x12+15x72−5x12f\left(x\right)=\frac{7x^{\frac{7}{2}}+x^{\frac{1}{2}}+1}{5x^{\frac{7}{2}}-5x^{\frac{1}{2}}}?
Asymptotes of Exponential
Determine if the function g(x)=e−x2+2x−3g\left(x\right)=e^{-x^2+2x-3} has any horizontal asymptotes.
Limits at infinity: Indeterminate forms
Which of the following statements is true about the functionf(x)=7x72+x12+15x72−5x12f\left(x\right)=\frac{7x^{\frac{7}{2}}+x^{\frac{1}{2}}+1}{5x^{\frac{7}{2}}-5x^{\frac{1}{2}}}?
Trigonometric Limits
Practice: Trig Limits
Evaluate lim⁡x→0 2sin⁡5xx2\displaystyle\lim_{x\to0}\ \frac{2\sin5x}{x^2}.
Infinite Limits: Evaluating limits
Evaluate lim⁡ x→2+x−32−x\lim\ _{x\rightarrow2^+}\frac{x-3}{2-x}
Infinite Limits: Evaluating limits
Evaluate lim⁡ x→2+x−32−x\lim\ _{x\rightarrow2^+}\frac{x-3}{2-x}
Trigonometric Limits
Practice: Trig Limits
Evaluate lim⁡x→0 2sin⁡5xx2\displaystyle\lim_{x\to0}\ \frac{2\sin5x}{x^2}.
Practice: Log Limits
Determine lim⁡x →0 ln⁡(−ln⁡(x2))\lim_{x\ \rightarrow0}\ \ln\left(-\ln\left(x^2\right)\right).
Infinite Limits
Q:\bf{Q:} Determine if the function f(x)=eπ+1ln⁡(3x−2)f\left(x\right)=\displaystyle\frac{e^{\pi}+1}{\ln\left(3x-2\right)} has any vertical asymptotes.
Infinite Limits
Determine lim⁡x→π−ln⁡(sin⁡x)\displaystyle\lim_{x \rightarrow\pi^-}\ln(\sin x) .
Find all asymptotes for f(x)=3x2+x+1x2−1\displaystyle f(x)=\frac{3x^2+x+1}{x^2-1} . Enter as comma-separated values (no spaces in between), smaller value first
Find the exact list of vertical asymptotes of f(x)=x2+4xx2−16\displaystyle f(x)=\frac{x^2+4x}{x^2-16}.
Asymptotes of Exponential
Determine if the function g(x)=e−x2+2x−3g\left(x\right)=e^{-x^2+2x-3} has any horizontal asymptotes.
General Indeterminate Forms
Evaluate:
lim⁡x→+∞2x+19x2−2x−6\lim_{x \rightarrow +\infty} \frac{2x + 1}{\sqrt{9x^2-2x - 6}}
Infinite Limits and Continuity
Given the function f(x)={arctan⁡x+π/2if x≤02xif 0<x≤e1ln⁡xif x>ef(x)=\begin{cases} \arctan x+\pi/2&\text{if }x \le 0\\ \frac{2}{x}&\text{if }0<x\le e\\ \frac{1}{\ln x}&\text{if }x>e \end{cases}, which of the following statements is true about f(x)f(x)?
Limits at infinity: Indeterminate forms
Which of the following statements is true about the functionf(x)=7x72+x12+15x72−5x12f\left(x\right)=\frac{7x^{\frac{7}{2}}+x^{\frac{1}{2}}+1}{5x^{\frac{7}{2}}-5x^{\frac{1}{2}}}?
Limits at Infinity
lim⁡x→−∞x2+10x−8x+56x−4\displaystyle\lim_{x\rightarrow-\infty}\frac{\sqrt{x^2+10x}-8x+5}{6x-4}
Infinite Limits
lim⁡x→2(x−9)(x+2)x−2\displaystyle \lim_{x\rightarrow2}\frac{(x-9)(x+2)}{x-2}
Limits at Infinity
Find the asymptote(s), if any, of
y=x3−1x2y=\dfrac{x^3 -1}{x^2}
Determine the value of lim⁡t→2+log⁡3(1t−2)\displaystyle \lim_{t \to 2^+}\log_3 \left( \frac{1}{t-2} \right).
Infinite Limits
Q:\bf{Q:} Determine if the function f(x)=eπ+1ln⁡(3x−2)f\left(x\right)=\displaystyle\frac{e^{\pi}+1}{\ln\left(3x-2\right)} has any vertical asymptotes.
Asymptotes of Exponential
Determine if the function g(x)=e−x2+2x−3g\left(x\right)=e^{-x^2+2x-3} has any horizontal asymptotes.
Infinite Limits
Determine lim⁡x→π−ln⁡(sin⁡x)\displaystyle\lim_{x \rightarrow\pi^-}\ln(\sin x) .
Find all asymptotes for f(x)=3x2+x+1x2−1\displaystyle f(x)=\frac{3x^2+x+1}{x^2-1} . Enter as comma-separated values (no spaces in between), smaller value first
Find the exact list of vertical asymptotes of f(x)=x2+4xx2−16\displaystyle f(x)=\frac{x^2+4x}{x^2-16}.
Determine if the function h(t)=eπ+1ln⁡(3x−2)h\left(t\right)=\frac{e^{\pi}+1}{\ln\left(3x-2\right)} has any vertical asymptotes.
Infinite Limits
Find the exact list of vertical asymptotes of
f(x)=x2+4xx2−16f(x)=\frac{x^2+4x}{x^2-16}
Limits: Practice
Find the following limit:
lim⁡x→2−5(x+1)x−2\displaystyle\lim_{x\rightarrow 2}\frac{-5(x+1)}{x-2}
🌶 TOUGH! Evaluate the limit, lim⁡x→2−5(x+1)x−2\displaystyle\lim_{x\rightarrow 2}\frac{-5(x+1)}{x-2}.
Determine if the function h(x)=eπ+1ln⁡(3x−2)h\left(x\right)=\frac{e^{\pi}+1}{\ln\left(3x-2\right)} has any vertical asymptotes.
🦊 TRICKY! Evaluate the limit, lim⁡x→∞x2−10x−x2\displaystyle\lim_{x\rightarrow \infty} \sqrt{x^2-10x}-x^2.
Infinite Limits
Evaluate lim⁡x→∞arctan⁡(x−x3)\displaystyle \lim_{x\to\infty}\arctan\left(x-x^3\right)
Infinite Limits: Evaluating limits
Evaluate lim⁡ x→2+x−32−x\lim\ _{x\rightarrow2^+}\frac{x-3}{2-x}
Practice: Log Limits
Determine lim⁡x →0 ln⁡(−ln⁡(x2))\lim_{x\ \rightarrow0}\ \ln\left(-\ln\left(x^2\right)\right).
Trigonometric Limits
Practice: Trig Limits
Evaluate lim⁡x→0 2sin⁡5xx2\displaystyle\lim_{x\to0}\ \frac{2\sin5x}{x^2}.