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Calculus 2 : Finding Limits and One-Sided Limits

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Example Questions

Example Question #161 : Limits

Evaluate the limit:

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

Possible Answers:

$-\frac{1}{3}$

DNE

$\infty$

Correct answer:

$-\frac{1}{3}$

Explanation:

The limiting situation in this equation would be the denominator. Plug the value that x is approaching into the denominator to see if the denominator will equal 0. In this question, the denominator will equal zero when x=-1; so we try to eliminate the denominator by factoring.

When the denominator is no longer zero, we may continue to insert the value of x into the remaining equation.

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

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Example Question #121 : Finding Limits And One Sided Limits

Evaluate the limit:

$\lim_{x \to 2}\frac{x^2+5x-14}{x^2-5x+6}$

Possible Answers:

$\infty$

DNE

Correct answer:

Explanation:

The limiting situation in this equation would be the denominator. Plug the value that x is approaching into the denominator to see if the denominator will equal 0. In this question, the denominator will equal zero when x=2; so we try to eliminate the denominator by factoring.

When the denominator is no longer zero, we may continue to insert the value of x into the remaining equation.

$\\ \lim_{x \to 2}\frac{x^2+5x-14}{x^2-5x+6}\\ \\=\lim_{x \to 2}\frac{(x-2)(x+7)}{(x-2)(x-3)}\\ \\=\lim_{x \to 2}\frac{x+7}{x-3}\\ \\=\frac{2+7}{2-3}\\ \\=\frac{9}{-1}\\ \\=-9$

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Example Question #122 : Finding Limits And One Sided Limits

Evaluate the limit:

$\lim_{x \to 3}\frac{x^2-6x+9}{x^4-81}$

Possible Answers:

$\infty$

$-\infty$

DNE

Correct answer:

Explanation:

The limiting situation in this equation would be the denominator. Plug the value that x is approaching into the denominator to see if the denominator will equal 0. In this question, the denominator will equal zero when x=3; so we try to eliminate the denominator by factoring.

When the denominator is no longer zero, we may continue to insert the value of x into the remaining equation.

$\\ \lim_{x \to 3}\frac{x^2-6x+9}{x^4-81}\\ \\=\lim_{x \to 3}\frac{(x-3)(x-3)}{(x^2+9)(x^2-9)}\\ \\=\lim_{x \to 3}\frac{(x-3)(x-3)}{(x^2+9)(x+3)(x-3)}\\ \\=\lim_{x \to 3}\frac{x-3}{(x^2+9)(x+3)}\\ \\=\frac{3-3}{(3^2+9)(3+3)}\\ \\=\frac{0}{(9+9)(6)}\\ \\=\frac{0}{18*6}\\ \\=\frac{0}{108}=0$

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Example Question #121 : Finding Limits And One Sided Limits

Evaluate the limit:

$\lim_{x \to 9}\frac{x^2-9x}{x^2-8x-9}$

Possible Answers:

$\infty$

DNE

$\frac{9}{10}$

Correct answer:

$\frac{9}{10}$

Explanation:

The limiting situation in this equation would be the denominator. Plug the value that x is approaching into the denominator to see if the denominator will equal 0. In this question, the denominator will equal zero when x=9; so we try to eliminate the denominator by factoring.

When the denominator is no longer zero, we may continue to insert the value of x into the remaining equation.

$\\\lim_{x \to 9}\frac{x^2-9x}{x^2-8x-9}\\ \\=\lim_{x \to 9}\frac{x(x-9)}{(x-9)(x+1)}\\ \\=\lim_{x \to 9}\frac{x}{x+1}\\ \\=\frac{9}{9+1}=\frac{9}{10}$

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Example Question #124 : Finding Limits And One Sided Limits

Evaluate the limit:

$\lim_{x \to -3}\frac{x^2-9}{x^2+7x+12}$

Possible Answers:

DNE

$\infty$

Correct answer:

Explanation:

The limiting situation in this equation would be the denominator. Plug the value that x is approaching into the denominator to see if the denominator will equal 0. In this question, the denominator will equal zero when x=-3; so we try to eliminate the denominator by factoring.

When the denominator is no longer zero, we may continue to insert the value of x into the remaining equation.

$\\\lim_{x \to -3}\frac{x^2-9}{x^2+7x+12}\\ \\=\lim_{x \to -3}\frac{(x+3)(x-3)}{(x+3)(x+4)}\\ \\=\lim_{x \to -3}\frac{x-3}{x+4}\\ \\=\frac{-3-3}{-3+4}\\ \\=\frac{-6}{1}\\ \\=-6$

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Example Question #122 : Finding Limits And One Sided Limits

Evaluate the limit:

$\lim_{x \to 0}\frac{x^3+x^2}{x^4+x^3}$

Possible Answers:

$\infty$

$-\infty$

DNE

Correct answer:

DNE

Explanation:

The limiting situation in this equation would be the denominator. Plug the value that x is approaching into the denominator to see if the denominator will equal 0. In this question, the denominator will equal zero when x=0; so we try to eliminate the denominator by factoring.

When the denominator is no longer zero, we may continue to insert the value of x into the remaining equation. We see that we can no longer factor this to make the denominator not equal 0; hence this limit DNE because the denominator is zero.

$\\ \lim_{x \to 0}\frac{x^3+x^2}{x^4+x^3}\\ \\=\lim_{x \to 0}\frac{x^2(x+1)}{x^3(x+1)}\\ \\=\lim_{x \to 0}\frac{1}{x}\\ \\=\frac{1}{0}\Rightarrow DNE$

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Example Question #122 : Finding Limits And One Sided Limits

Evaluate the limit:

$\lim_{x \to 2}\frac{x-2}{x+2}$

Possible Answers:

$-\infty$

DNE

$\infty$

Correct answer:

Explanation:

The limiting situation in this equation would be the denominator. Plug the value that x is approaching into the denominator to see if the denominator will equal 0. In this question, the denominator will not equal zero when x=2; so we proceed to insert the value of x into the entire equation.

$\lim_{x \to 2}\frac{x-2}{x+2}=\frac{2-2}{2+2}=\frac{0}{4}=0$

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Example Question #171 : Calculus Ii

Evaluate the limit:

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

Possible Answers:

$-\infty$

DNE

$\infty$

Correct answer:

DNE

Explanation:

The limiting situation in this equation would be the denominator. Plug the value that x is approaching into the denominator to see if the denominator will equal 0. In this question, the denominator will equal zero when x=1; so we try to eliminate the denominator by factoring.

$\\ \lim_{x \to 1}\frac{3x^2+3x}{x^2-1}\\ \\=\lim_{x \to 1}\frac{3x(x+1)}{(x-1)(x+1)}\\ \\=\lim_{x \to 1}\frac{3x}{x-1}\\ \\=\frac{3}{0}\Rightarrow DNE$

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Example Question #129 : Finding Limits And One Sided Limits

Screen shot 2015 07 27 at 12.31.44 pm

Given the above graph of f(x) , what is $\lim_{x\rightarrow 0}f(x)$ ?

Possible Answers:

Does Not Exist

$\infty$

$-\infty$

Correct answer:

Does Not Exist

Explanation:

Examining the graph, we can observe that $\lim_{x\rightarrow 0}f(x)$ does not exist, as f(x) is not continuous at x=0 . We can see this by checking the three conditions for which a function f(x) is continuous at a point (c, f(c)) :

A value exists in the domain of
The limit of exists as approaches
The limit of at is equal to

Given c=0 , we can see that condition #1 is not satisfied because the graph has a vertical asymptote instead of only one value for f(0) and is therefore an infinite discontinuity at (0,f(0)) .

We can also see that condition #2 is not satisfied because $\lim_{x\rightarrow 0}f(x)$ approaches two different limits: $-\infty$ from the left and $\infty$ from the right.

Based on the above, condition #3 is also not satisfied because $\lim_{x\rightarrow 0}f(x)$ is not equal to the multiple values of f(c) .

Thus, $\lim_{x\rightarrow 0}f(x)$ does not exist.

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Example Question #130 : Finding Limits And One Sided Limits

Screen shot 2015 07 28 at 6.24.36 pm

Given the above graph of f(x) , what is $\lim_{x\rightarrow 0}f(x)$ ?

Possible Answers:

$-\infty$

$\infty$

Does Not Exist

Correct answer:

Does Not Exist

Explanation:

A value exists in the domain of
The limit of exists as approaches
The limit of at is equal to

Given c=0 , we can see that condition #1 is not satisfied because the graph has a vertical asymptote instead of only one value for f(0) and is therefore an infinite discontinuity at (0,f(0)) .

We can also see that condition #2 is not satisfied because $\lim_{x\rightarrow 0}f(x)$ approaches two different limits: $\infty$ from the left and $-\infty$ from the right.

Based on the above, condition #3 is also not satisfied because $\lim_{x\rightarrow 0}f(x)$ is not equal to the multiple values of f(c) .

Thus, $\lim_{x\rightarrow 0}f(x)$ does not exist.

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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 #161 : Limits

Example Question #121 : Finding Limits And One Sided Limits

Example Question #122 : Finding Limits And One Sided Limits

Example Question #121 : Finding Limits And One Sided Limits

Example Question #124 : Finding Limits And One Sided Limits

Example Question #122 : Finding Limits And One Sided Limits

Example Question #122 : Finding Limits And One Sided Limits

Example Question #171 : Calculus Ii

Example Question #129 : Finding Limits And One Sided Limits

Example Question #130 : Finding Limits And One Sided Limits

All Calculus 2 Resources

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