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General Indeterminate Forms

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Wize University Calculus 1 Textbook > Applications of Differentiation

Solving General Indeterminate Forms

4 Activities

Wize University Calculus 1 Textbook > Limits

Infinite Limits

3 Activities

Evaluate:

lim⁡x→+∞2x+19x2−2x−6\lim_{x \rightarrow +\infty} \frac{2x + 1}{\sqrt{9x^2-2x - 6}}


More Solving General Indeterminate Forms Questions:
L'Hopital's Rule: General Indeterminate Forms
Evaluate the limit L=lim⁡x→0+xsin⁡ xL=\lim\limits_{x\rightarrow0^+}x^{\sin\ x}.
L'Hospital's Rule: Indeterminate Forms
Evaluate lim⁡ x→∞(1−kx)3x\lim\ _{x\rightarrow\infty}\left(1-\frac{k}{x}\right)^{3x}.
Indeterminate Forms
Evaluate the limit if it exists.
lim⁡x→∞ (1+1x)x\lim_{x\to\infty}\ \left(1+\frac{1}{x}\right)^x.
L'Hospital's Rule: Indeterminate Forms
Evaluate lim⁡ x→∞(1−kx)3x\lim\ _{x\rightarrow\infty}\left(1-\frac{k}{x}\right)^{3x}.
L'Hospital's Rule: Indeterminate Forms
Find lim⁡x→0+(ln⁡x)(sin⁡x)\lim_{x\rightarrow0^+}\left(\ln x\right)\left(\sin x\right).
L'Hospital's Rule: Indeterminate Forms
Evaluate lim⁡x→0+(e1x)1ln⁡x\lim_{x\rightarrow0^+}\left(e^{\frac{1}{x}}\right)^{^{\frac{1}{\ln x}}}.
Indeterminate Forms
Evaluate the limit: lim⁡x→0+(2x)x2\displaystyle \lim_{x\rightarrow 0^+}(2x)^{\frac{x}{2}}
Indeterminate Forms: Polynomial and Ln
Evaluate the limit: lim⁡x→0+xaln⁡x\displaystyle \lim_{x\rightarrow 0^+}x^a\ln{x}, where a>0a>0.
Practice: Fractions with Trig
Q.\textbf{Q.} Evaluate: lim⁡x→0+(2x−2sin⁡x)\displaystyle \lim_{x\rightarrow 0^+}\left(\frac{2}{x}-\frac{2}{\sin{x}}\right)
Evaluate the following limit: lim⁡x→0+(2x)x2\displaystyle \lim_{x\rightarrow 0^+}(2x)^{\frac{x}{2}}
Evaluate the following limit: lim⁡x→0+x32ln⁡x\displaystyle \lim_{x\rightarrow0^+}x^{\frac{3}{2}}\ln{x}
Practice: $f^g$ Form with Trig
Q.\textbf{Q.} Evaluate the following limit: lim⁡x→0+cos⁡xcot⁡x\displaystyle \lim_{x\rightarrow 0^+} \cos{x}^{\cot{x}}
Indeterminate Forms
Evaluate the limit: lim⁡x→∞[1+sin⁡(2x)]2x\displaystyle \lim_{x\rightarrow\infty}\left[1+\sin\left(\frac{2}{x}\right)\right]^{2x}
Practice: L'Hospital's Rule
Evaluate lim⁡x→0+(e1x)1ln⁡x\lim_{x\rightarrow0^+}\left(e^{\frac{1}{x}}\right)^{^{\frac{1}{\ln x}}}.
Practice: L'Hospital's Rule
Evaluate lim⁡x→∞ x3 sin⁡(1x2)\displaystyle \lim_{x\to\infty}\ x^3\ \sin\left(\frac{1}{x^2}\right).
Practice: L'Hospital's Rule
Evaluate lim⁡x→∞ x3 sin⁡(1x2)\displaystyle \lim_{x\to\infty}\ x^3\ \sin\left(\frac{1}{x^2}\right).
Practice: L'Hospital's Rule
Evaluate lim⁡x→0+(e1x)1ln⁡x\lim_{x\rightarrow0^+}\left(e^{\frac{1}{x}}\right)^{^{\frac{1}{\ln x}}}.
Practice: L'Hospital's Rule
Evaluate lim⁡x→∞ x3 sin⁡(1x2)\displaystyle \lim_{x\to\infty}\ x^3\ \sin\left(\frac{1}{x^2}\right).
Evaluate lim⁡x→0+2ln⁡xln⁡(ex−1)\lim_{x\rightarrow0^+}\frac{2\ln x}{\ln\left(e^x-1\right)}
General Indeterminate Forms
Evaluate
lim⁡x→0+xaln⁡x where a>0\lim\limits_{x\to0^+} x^a\ln x \text{ where } a>0
l'Hopital's Rule: General Indeterminate Forms
Evaluate
lim⁡x→0+x3ln⁡x\lim\limits_{x\to0^+}x^3\ln x
Evaluate the following limit: lim⁡x→0+(2x)x2\displaystyle \lim_{x\rightarrow 0^+}(2x)^{\frac{x}{2}}
Evaluate the following limit: lim⁡x→0+x32ln⁡x\displaystyle \lim_{x\rightarrow0^+}x^{\frac{3}{2}}\ln{x}
Evaluate the following limit: lim⁡x→0+x32ln⁡x\displaystyle \lim_{x\rightarrow0^+}x^{\frac{3}{2}}\ln{x}
Evaluate the following limit: lim⁡x→0+(2x)x2\displaystyle \lim_{x\rightarrow 0^+}(2x)^{\frac{x}{2}}
Indeterminate Forms: Power and Ln
Evaluatelim⁡x→0+x32ln⁡x\displaystyle \lim_{x\rightarrow 0^+} x^{\frac{3}{2}}\ln{x}
Practice: $f^g$ Form with Trig
Q.\textbf{Q.} Evaluate the following limit: lim⁡x→0+cos⁡xcot⁡x\displaystyle \lim_{x\rightarrow 0^+} \cos{x}^{\cot{x}}
Practice: Fractions with Trig
Q.\textbf{Q.} Evaluate: lim⁡x→0+(2x−2sin⁡x)\displaystyle \lim_{x\rightarrow 0^+}\left(\frac{2}{x}-\frac{2}{\sin{x}}\right)
Indeterminate Forms
Evaluate the limit: lim⁡x→∞[1+sin⁡(2x)]2x\displaystyle \lim_{x\rightarrow\infty}\left[1+\sin\left(\frac{2}{x}\right)\right]^{2x}
Indeterminate Forms: Polynomial and Ln
Evaluate the limit: lim⁡x→0+xaln⁡x\displaystyle \lim_{x\rightarrow 0^+}x^a\ln{x}, where a>0a>0.
Indeterminate Forms
Evaluate the limit: lim⁡x→0+(2x)x2\displaystyle \lim_{x\rightarrow 0^+}(2x)^{\frac{x}{2}}
Determine lim⁡x→∞(1+7x)x5\lim_{x\rightarrow\infty}\left(1+\frac{7}{x}\right)^{\frac{x}{5}}.
Determine lim⁡ x→∞(x)1x2\lim\ _{x\rightarrow\infty}\left(\sqrt{x}\right)^{\frac{1}{x^2}}.
Evaluate the limit lim⁡x→∞(1+sin⁡2x)2x\displaystyle \lim_{x\rightarrow \infty}\left(1+\sin{\frac{2}{x}}\right)^{2x}.
Evaluate the limit: lim⁡x→0+(2x−2sin⁡x)\displaystyle \lim_{x\rightarrow 0^+}\left(\frac{2}{x}-\frac{2}{\sin{x}}\right)
Solve the limit lim⁡x →0(1−2x)1x2\displaystyle\lim_{x\ \to0}\left(1-2x\right)^{\frac{1}{x^2}}.
Find lim⁡x→0+(ln⁡x)(sin⁡x)\lim_{x\rightarrow0^+}\left(\ln x\right)\left(\sin x\right).
🦊 TRICKY! Determine lim⁡x→2+(x−2)1x−2\lim_{x\rightarrow2^+}\left(x-2\right)^{\frac{1}{x-2}}.
L'Hopital's Rule: General Indeterminate Forms
Evaluate the limit L=lim⁡x→0+xsin⁡ xL=\lim\limits_{x\rightarrow0^+}x^{\sin\ x}.
l'Hopital's Rule: General Indeterminate Forms
Evaluate
lim⁡x→∞(1+7x)x/5\lim_{x\to \infty}\left(1+\frac{7}{x}\right)^{x/5}
Evaluate lim⁡x→0+x32ln⁡x\lim_{x\rightarrow0^+}x^{\frac{3}{2}}\ln x
Determine lim⁡x→∞(1+7x)x5\lim_{x\rightarrow\infty}\left(1+\frac{7}{x}\right)^{\frac{x}{5}}
Indeterminate Forms
Evaluate the limit if it exists.
lim⁡x→∞ (1+1x)x\lim_{x\to\infty}\ \left(1+\frac{1}{x}\right)^x.
Indeterminate Forms
Evaluate the limit, if it exists.
lim⁡x→3 1x−3−e1x−3\lim_{x\to3}\ \frac{1}{x-3}-e^{\frac{1}{x-3}}.
Indeterminate Forms
Evaluate the limit, if it exists.
lim⁡x→−2 (x+2)ln⁡∣x+2∣\lim_{x\to-2}\ \left(x+2\right)\ln\left|x+2\right|
L'Hospital's Rule: Indeterminate Forms
Evaluate lim⁡ x→∞(1−kx)3x\lim\ _{x\rightarrow\infty}\left(1-\frac{k}{x}\right)^{3x}.
L'Hospital's Rule: Indeterminate Forms
Evaluate lim⁡x→0+(e1x)1ln⁡x\lim_{x\rightarrow0^+}\left(e^{\frac{1}{x}}\right)^{^{\frac{1}{\ln x}}}.
L'Hospital's Rule: Indeterminate Forms
Find lim⁡x→0+(ln⁡x)(sin⁡x)\lim_{x\rightarrow0^+}\left(\ln x\right)\left(\sin x\right).
Evaluate lim⁡x→0+x32ln⁡x\lim_{x\rightarrow0^+}x^{\frac{3}{2}}\ln x
Practice: L'hopital's rule
Q.\textbf{Q.} Evaluate the following limits.
1. lim⁡x→0+cos⁡xcot⁡x\displaystyle \lim_{x\rightarrow 0^+} \cos{x}^{\cot{x}}
2. lim⁡x→∞[x2+1−x]\displaystyle \lim_{x\rightarrow \infty}\left[\sqrt{x^2+1}-x\right]
More Infinite Limits Questions:
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.
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}.