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Calculus 2 : Finding Limits and One-Sided Limits

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Example Questions

Example Question #391 : Finding Limits And One Sided Limits

Evaluate the following limit:

$\lim_{x \to 0} \frac{2x^2}{5}$

Possible Answers:

$\frac{2}{5}$

$\infty$

$-\infty$

Correct answer:

Explanation:

The first step is to always plug in the value of the limit. Doing so we get

$\frac{2(0)^2)}{5}=0$

Since this is a valid answer, no further work is required.

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Example Question #392 : Finding Limits And One Sided Limits

Evaluate the following limit:

$\lim_{x \to 0} \frac{5x+10}{x+1}$

Possible Answers:

$\infty$

Correct answer:

Explanation:

The first step is to always plug in the value of the limit. Doing so we get

$\frac{5(0)+10}{0+1}=\frac{10}{1}=10$

Since this is a valid answer, no further work is required.

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Example Question #393 : Finding Limits And One Sided Limits

Evaluate the following limit:

$\lim_{x \to 2} \frac{3x^2 + 2x}{x-1}$

Possible Answers:

$\infty$

Correct answer:

Explanation:

The first step is to always plug in the value of the limit. Doing so we get

$\frac{3(2)^2+2(2)}{2-1}=\frac{(3)(4)+(2)(2)}{1}=16$

Since this is a valid answer, no further work is required.

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Example Question #394 : Finding Limits And One Sided Limits

Evaluate the following limit:

$\lim_{x \to 3} \frac{x^2-x-6}{x^2-9}$

Possible Answers:

$\textup{Indeterminate}$

$\frac{5}{6}$

$-\infty$

Correct answer:

$\frac{5}{6}$

Explanation:

The first step is to always plug in the value of the limit. Doing so we get

$\frac{(3^2-3-6)}{3^2-9}=\frac{0}{0}$

This is indeterminate. We need to verify it is indeterminate and therefore need take another approach. Let's take a look at the original equation again.

$\frac{x^2-x-6}{x^2-9}$

Taking what we learned in algebra, we know that both the numerator and denominator of the equation are factorable. The equation becomes

$\frac{(x-3)(x+2)}{(x-3)(x+3)}$

The (x-3) from the top and bottom cancels, which then simplifies to

$\frac{(x+2)}{(x+3)}$

Now we evaluate the limit once again using the simplified equation and we get

$\frac{(3+2)}{(3+3)}=\frac{5}{6}$

This is a valid answer.

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Example Question #395 : Finding Limits And One Sided Limits

$\lim_{x \to 0} \frac{(4x^2-2x)}{(x)}$

Possible Answers:

$\infty$

$\textup{Indeterminate}$

Correct answer:

Explanation:

The first step is to always plug in the value of the limit. Doing so we get

$\frac{(4(0)^2-2(0))}{0}=\frac{0}{0}$

This is indeterminate. We need to verify it is indeterminate and therefore need take another approach. Let's take a look at the original equation again.

$\frac{4x^2-2x}{x}$

Taking what we learned in algebra, we know that the numerator is factorable. Taking out an yields

$\frac{(x)(4x-2)}{(x)}$

The (x) from the top and bottom cancels, which then simplifies to

$\frac{(4x-2)}{(1)}$

Now we evaluate the limit once again using the simplified equation and we get

4(0)-2=-2

This is a valid answer

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Example Question #396 : Finding Limits And One Sided Limits

$\begin{align*}&\text{Evaluate the following limit:}\\&lim_{x\rightarrow1-}\frac{41\cdot (x - 1)^{2}}{46ln(x^{8})}\end{align*}$

Possible Answers:

$\frac{41}{135424}$

$-\frac{41}{135424}$

$\infty$

$-\infty$

Correct answer:

Explanation:

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Example Question #397 : Finding Limits And One Sided Limits

$\begin{align*}&\text{Evaluate the following limit:}\\&lim_{x\rightarrow1-}\frac{49ln(x^{8})}{48\cdot (x - 1)^{3}}\end{align*}$

Possible Answers:

$-\frac{(49\cdot (-48)^{(\frac{2}{3})})}{6}$

$-\infty$

$\infty$

$\frac{(49\cdot (-48)^{(\frac{2}{3})})}{6}$

Correct answer:

$\infty$

Explanation:

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Example Question #394 : Finding Limits And One Sided 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:

$\infty$

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

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

$-\infty$

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 #395 : Finding Limits And One Sided Limits

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

Possible Answers:

$\infty$

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

$-\infty$

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

Correct answer:

$\infty$

Explanation:

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Example Question #395 : Finding Limits And One Sided Limits

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

Possible Answers:

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

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

$\infty$

$-\infty$

Correct answer:

Explanation:

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Calculus 2 : Finding Limits and One-Sided Limits

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #391 : Finding Limits And One Sided Limits

Example Question #392 : Finding Limits And One Sided Limits

Example Question #393 : Finding Limits And One Sided Limits

Example Question #394 : Finding Limits And One Sided Limits

Example Question #395 : Finding Limits And One Sided Limits

Example Question #396 : Finding Limits And One Sided Limits

Example Question #397 : Finding Limits And One Sided Limits

Example Question #394 : Finding Limits And One Sided Limits

Example Question #395 : Finding Limits And One Sided Limits

Example Question #395 : Finding Limits And One Sided Limits

All Calculus 2 Resources

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