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Calculus 2 : Limits

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Example Questions

Example Question #441 : Limits

$\begin{align*}&\text{Evaluate the following limit:}\\&lim_{x\rightarrow1+}\frac{43sin(15\cdot \pi x)^{3}}{47cos(\frac{(\pi x)}{2})^{2}}\end{align*}$

Possible Answers:

$-\frac{(145125\cdot \pi ^{3})}{(\frac{(47\cdot \pi ^{2})}{4})^{(\frac{3}{2})}}$

$\infty$

$\frac{(145125\cdot \pi ^{3})}{(\frac{(47\cdot \pi ^{2})}{4})^{(\frac{3}{2})}}$

$-\infty$

Correct answer:

Explanation:

$\begin{align*}&\text{Examining this problem, we find that both}\\&\text{numerator and denominator have zero values at the limit specified.}\\&\text{Noting this, L'Hopital's method may be of use:}\\&\text{If both numerator and denominator go to }0\text{ or }\pm\infty\\&lim_{x\rightarrow n}\frac{f(x)}{g(x)}=lim_{x\rightarrow n}\frac{f'(x)}{g'(x)}\\&\frac{43sin(15\cdot \pi x)^{3}}{47cos(\frac{(\pi x)}{2})^{2}}\rightarrow\frac{0}{0}\\&\frac{1935\cdot \pi cos(15\cdot \pi x)sin(15\cdot \pi x)^{2}}{-47\cdot \pi cos(\frac{(\pi x)}{2})sin(\frac{(\pi x)}{2})}\rightarrow\frac{0}{0}\\&\frac{58050\cdot \pi ^{2}cos(15\cdot \pi x)^{2}sin(15\cdot \pi x) - 29025\cdot \pi ^{2}sin(15\cdot \pi x)^{3}}{\frac{(47\cdot \pi ^{2}sin(\frac{(\pi x)}{2})^{2})}{2}-\frac{ (47\cdot \pi ^{2}cos(\frac{(\pi x)}{2})^{2})}{2}}\rightarrow\frac{0}{\frac{(47\cdot \pi ^{2})}{2}}=0\\&lim_{x\rightarrow1+}\frac{43sin(15\cdot \pi x)^{3}}{47cos(\frac{(\pi x)}{2})^{2}}=0\end{align*}$

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Example Question #441 : Calculus Ii

$\begin{align*}&\text{Evaluate the following limit:}\\&lim_{x\rightarrow1-}\frac{-6ln(x^{7})}{5sin(10\cdot \pi x)^{2}}\end{align*}$

Possible Answers:

$-\frac{42}{(500\cdot \pi ^{2})^{(\frac{1}{2})}}$

$\infty$

$\frac{42}{(500\cdot \pi ^{2})^{(\frac{1}{2})}}$

$-\infty$

Correct answer:

$\infty$

Explanation:

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Example Question #443 : Limits

$\begin{align*}&\text{Evaluate the following limit:}\\&lim_{x\rightarrow1-}\frac{18tan(20\cdot \pi x)^{2}}{19x - 19}\end{align*}$

Possible Answers:

$\frac{(7200\cdot \pi ^{2})}{361}$

$\infty$

$-\frac{(7200\cdot \pi ^{2})}{361}$

$-\infty$

Correct answer:

Explanation:

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Example Question #401 : Finding Limits And One Sided Limits

$\begin{align*}&\text{Evaluate the following limit:}\\&lim_{x\rightarrow1+}\frac{-38\cdot (x - 1)^{3}}{13sin(3\cdot \pi x)^{3}}\end{align*}$

Possible Answers:

$\infty$

$-\infty$

$\frac{38}{(351\cdot \pi ^{3})}$

$-\frac{38}{(351\cdot \pi ^{3})}$

Correct answer:

$\frac{38}{(351\cdot \pi ^{3})}$

Explanation:

$\begin{align*}&\text{Examining this problem, we find that both}\\&\text{numerator and denominator have zero values at the limit specified.}\\&\text{Noting this, L'Hopital's method may be of use:}\\&\text{If both numerator and denominator go to }0\text{ or }\pm\infty\\&lim_{x\rightarrow n}\frac{f(x)}{g(x)}=lim_{x\rightarrow n}\frac{f'(x)}{g'(x)}\\&\frac{-38\cdot (x - 1)^{3}}{13sin(3\cdot \pi x)^{3}}\rightarrow\frac{0}{0}\\&\frac{-114\cdot (x - 1)^{2}}{117\cdot \pi cos(3\cdot \pi x)sin(3\cdot \pi x)^{2}}\rightarrow\frac{0}{0}\\&\frac{228 - 228x}{702\cdot \pi ^{2}cos(3\cdot \pi x)^{2}sin(3\cdot \pi x) - 351\cdot \pi ^{2}sin(3\cdot \pi x)^{3}}\rightarrow\frac{0}{0}\\&\frac{-228}{2106\cdot \pi ^{3}cos(3\cdot \pi x)^{3} - 7371\cdot \pi ^{3}cos(3\cdot \pi x)sin(3\cdot \pi x)^{2}}\rightarrow\frac{-228}{-2106\cdot \pi ^{3}}\\&lim_{x\rightarrow1+}\frac{-38\cdot (x - 1)^{3}}{13sin(3\cdot \pi x)^{3}}=\frac{38}{(351\cdot \pi ^{3})}\end{align*}$

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Example Question #441 : Limits

$\begin{align*}&\text{Evaluate the following limit:}\\&lim_{x\rightarrow1-}\frac{-13\cdot (x - 1)^{2}}{-41cos(\frac{(7\cdot \pi x)}{2})}\end{align*}$

Possible Answers:

$-\infty$

$-\frac{52}{(82369\cdot \pi ^{2})}$

$\frac{52}{(82369\cdot \pi ^{2})}$

$\infty$

Correct answer:

Explanation:

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Example Question #442 : Limits

$\begin{align*}&\text{Evaluate the following limit:}\\&lim_{x\rightarrow0-}\frac{7tan(11x)^{2}}{29x^{3}}\end{align*}$

Possible Answers:

$\frac{(847\cdot (-29)^{(\frac{1}{3})})}{29}$

$\infty$

$-\infty$

$-\frac{(847\cdot (-29)^{(\frac{1}{3})})}{29}$

Correct answer:

$-\infty$

Explanation:

$\begin{align*}&\text{Examining this problem, we find that both}\\&\text{numerator and denominator have zero values at the limit specified.}\\&\text{Noting this, L'Hopital's method may be of use:}\\&\text{If both numerator and denominator go to }0\text{ or }\pm\infty\\&lim_{x\rightarrow n}\frac{f(x)}{g(x)}=lim_{x\rightarrow n}\frac{f'(x)}{g'(x)}\\&\frac{7tan(11x)^{2}}{29x^{3}}\rightarrow\frac{0}{0}\\&\frac{14tan(11x)\cdot (11tan(11x)^{2} + 11)}{87x^{2}}\rightarrow\frac{0}{0}\\&\frac{308tan(11x)^{2}\cdot (11tan(11x)^{2} + 11) + 14\cdot (11tan(11x)^{2} + 11)^{2}}{174x}\rightarrow\frac{1694}{0}=-\infty\\&lim_{x\rightarrow0-}\frac{7tan(11x)^{2}}{29x^{3}}=-\infty\end{align*}$

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Example Question #443 : Limits

$\begin{align*}&\text{Evaluate the following limit:}\\&lim_{x\rightarrow0+}\frac{33sin(17\cdot \pi x)}{35ln(x + 1)^{3}}\end{align*}$

Possible Answers:

$\frac{(561\cdot 35^{(\frac{2}{3})}\cdot \pi )}{35}$

$\infty$

$-\infty$

$-\frac{(561\cdot 35^{(\frac{2}{3})}\cdot \pi )}{35}$

Correct answer:

$\infty$

Explanation:

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Example Question #401 : Finding Limits And One Sided Limits

$\begin{align*}&\text{Evaluate the following limit:}\\&lim_{x\rightarrow0-}\frac{23x}{-5sin(18\cdot \pi x)}\end{align*}$

Possible Answers:

$-\frac{23}{(90\cdot \pi )}$

$\infty$

$-\infty$

$\frac{23}{(90\cdot \pi )}$

Correct answer:

$-\frac{23}{(90\cdot \pi )}$

Explanation:

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Example Question #406 : Finding Limits And One Sided Limits

$\begin{align*}&\text{Evaluate the following limit:}\\&lim_{x\rightarrow0-}\frac{26sin(19\cdot \pi x)}{21cos(5\cdot \pi x -\frac{ \pi }{2})^{2}}\end{align*}$

Possible Answers:

$\infty$

$-\infty$

$\frac{(494\cdot \pi )}{(525\cdot \pi ^{2})^{(\frac{1}{3})}}$

$-\frac{(494\cdot \pi )}{(525\cdot \pi ^{2})^{(\frac{1}{3})}}$

Correct answer:

$-\infty$

Explanation:

$\begin{align*}&\text{Examining this problem, we find that both}\\&\text{numerator and denominator have zero values at the limit specified.}\\&\text{Noting this, L'Hopital's method may be of use:}\\&\text{If both numerator and denominator go to }0\text{ or }\pm\infty\\&lim_{x\rightarrow n}\frac{f(x)}{g(x)}=lim_{x\rightarrow n}\frac{f'(x)}{g'(x)}\\&\frac{26sin(19\cdot \pi x)}{21cos(5\cdot \pi x -\frac{ \pi }{2})^{2}}\rightarrow\frac{0}{0}\\&\frac{494\cdot \pi cos(19\cdot \pi x)}{-210\cdot \pi cos(5\cdot \pi x -\frac{ \pi }{2})sin(5\cdot \pi x -\frac{ \pi }{2})}\rightarrow\frac{494\cdot \pi}{0}=-\infty\\&lim_{x\rightarrow0-}\frac{26sin(19\cdot \pi x)}{21cos(5\cdot \pi x -\frac{ \pi }{2})^{2}}=-\infty\end{align*}$

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Example Question #407 : Finding Limits And One Sided Limits

$\begin{align*}&\text{Evaluate the following limit:}\\&lim_{x\rightarrow1-}\frac{46sin(19\cdot \pi x)^{2}}{19ln(x^{5})}\end{align*}$

Possible Answers:

$\frac{(46\cdot \pi ^{2})}{25}$

$\infty$

$-\frac{(46\cdot \pi ^{2})}{25}$

$-\infty$

Correct answer:

Explanation:

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Calculus 2 : Limits

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #441 : Limits

Example Question #441 : Calculus Ii

Example Question #443 : Limits

Example Question #401 : Finding Limits And One Sided Limits

Example Question #441 : Limits

Example Question #442 : Limits

Example Question #443 : Limits

Example Question #401 : Finding Limits And One Sided Limits

Example Question #406 : Finding Limits And One Sided Limits

Example Question #407 : Finding Limits And One Sided Limits

All Calculus 2 Resources

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