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Calculus 2 : Finding Limits and One-Sided Limits

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Example Questions

Example Question #281 : Finding Limits And One Sided Limits

Evaluate the following limit:

$\lim_{x\rightarrow \infty} \frac{5x^4}{25x^4+3x^2+1}$

Possible Answers:

$\frac{1}{5}$

$\infty$

Correct answer:

$\frac{1}{5}$

Explanation:

To evaluate the limit as x approaches infinity, we must factor out a term consisting of the highest power term divided by itself (so we are unchanging the limit):

$\lim_{x\rightarrow \infty} (\frac{x^4}{x^4}) (\frac{5}{25+3x^{-1}+x^{-4}})$

The first term cancels to become 1, and the negative exponent terms go to zero when x approaches infinity (infinity in the denominator makes zero), so what we are left with is

$\frac{5}{25}=\frac{1}{5}$

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Example Question #282 : Finding Limits And One Sided Limits

Evaluate the limit:

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

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=1; so we proceed to insert the value of x into the entire equation.

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

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Example Question #283 : Finding Limits And One Sided Limits

Evaluate the limit:

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

Possible Answers:

DNE

$\infty$

$-\infty$

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.

$\lim_{x \to 2}\frac{2x+4}{x-4}=\frac{2*2+4}{2-4}=\frac{4+4}{-2}=-\frac{8}{2}=-4$

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Example Question #281 : Finding Limits And One Sided Limits

Evaluate the limit:

$\lim_{x \to 4}\frac{x+4}{x-4}$

Possible Answers:

$\infty$

DNE

$-\infty$

Correct answer:

DNE

Explanation:

We see that we cannot factor this to make the denominator not equal 0; hence this limit DNE because the denominator is zero.

$\lim_{x \to 4}\frac{x+4}{x-4}=\frac{4+4}{4-4}=\frac{8}{0}\Rightarrow DNE$

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Example Question #285 : Finding Limits And One Sided Limits

Evaluate the limit:

$\lim_{x \to 5}\frac{2x+5}{x+5}$

Possible Answers:

$\infty$

DNE

$-\infty$

1.5

Correct answer:

1.5

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=5; so we proceed to insert the value of x into the entire equation.

$\lim_{x \to 5}\frac{2x+5}{x+5}=\frac{2*5+5}{5+5}=\frac{10+5}{10}=\frac{15}{10}=1.5$

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Example Question #282 : Finding Limits And One Sided Limits

Evaluate the limit:

$\lim_{x \to 6}\frac{x^2-36}{6+x}$

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 not equal zero when x=6; so we proceed to insert the value of x into the entire equation.

$\lim_{x \to 6}\frac{x^2-36}{6+x}=\frac{6^2-36}{6+6}=\frac{36-36}{12}=\frac{0}{12}=0$

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Example Question #282 : Finding Limits And One Sided Limits

Evaluate the limit:

$\lim_{x \to 7}\frac{5x-7}{2x}$

Possible Answers:

DNE

$\infty$

$-\infty$

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=7; so we proceed to insert the value of x into the entire equation.

$\lim_{x \to 7}\frac{5x-7}{2x}=\frac{5*7-7}{2*7}=\frac{35-7}{14}=\frac{28}{14}=2$

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Example Question #282 : Finding Limits And One Sided Limits

Evaluate the limit:

$\lim_{x \to 8}\sqrt{x^2+2x+1}$

Possible Answers:

$\infty$

$-\infty$

DNE

Correct answer:

Explanation:

There is no limiting situation in this equation (like a denominator) so we can just plug in the value that n approaches into the limit and solve:

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

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Example Question #283 : Finding Limits And One Sided Limits

Evaluate the limit:

$\lim_{x \to 9}5x-10+3x^2$

Possible Answers:

$-\infty$

278

$\infty$

DNE

Correct answer:

278

Explanation:

There is no limiting situation in this equation (like a denominator) so we can just plug in the value that n approaches into the limit and solve:

$\lim_{x \to 9}5x-10+3x^2=5*9-10+3*9^2=45-10+243=35+243=278$

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Example Question #290 : Finding Limits And One Sided Limits

Evaluate the limit:

$\lim_{x \to 10}\frac{x^2-100}{x+10}$

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 not equal zero when x=10; so we proceed to insert the value of x into the entire equation.

$\lim_{x \to 10}\frac{x^2-100}{x+10}=\frac{10^2-100}{10+10}=\frac{100-100}{20}=\frac{0}{20}=0$

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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 #281 : Finding Limits And One Sided Limits

Example Question #282 : Finding Limits And One Sided Limits

Example Question #283 : Finding Limits And One Sided Limits

Example Question #281 : Finding Limits And One Sided Limits

Example Question #285 : Finding Limits And One Sided Limits

Example Question #282 : Finding Limits And One Sided Limits

Example Question #282 : Finding Limits And One Sided Limits

Example Question #282 : Finding Limits And One Sided Limits

Example Question #283 : Finding Limits And One Sided Limits

Example Question #290 : Finding Limits And One Sided Limits

All Calculus 2 Resources

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