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Calculus 2 : Finding Limits and One-Sided Limits

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Example Questions

Example Question #1 : Finding Limits And One Sided Limits

Which graph is a possible sketch of the function that possesses the following characteristics:

$f(-2)= undefined; \lim_{x \rightarrow -2^-} f(x)= \infty; \lim_{x \rightarrow -2^+} f(x)= -\infty;$

Possible Answers:

Graph2

Graph3

There does not exist such a graph.

Graph4

Graph1

Correct answer:

Graph1

Explanation:

Since $\tiny { f(-2) = undefined}$ there is a possible vertical asymptote at x=-2 .

As we approach x=-2 from the left, the graph should tend to $\tiny { +\infty}$ . Approaching x=-2 from the right, the graph tends to $\tiny {-\infty}$ .

The only graph that does so is Graph1 .

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Example Question #1 : Finding Limits And One Sided Limits

Rational_graph

The graph above is a sketch of the function . Find $\lim_{x \rightarrow 2^-}f(x)$ .

Possible Answers:

Does not exist

$-\infty$

$\infty$

Correct answer:

$-\infty$

Explanation:

We need to look at the behavior of the function as it tends to from the left. Therefore the answer is $\tiny { -\infty}$ .

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Example Question #2 : Finding Limits And One Sided Limits

Rational_graph

The graph above is a sketch of . Find $\lim_{x \rightarrow -2} f(x)$ .

Possible Answers:

Does not exist

$\infty$

$-\infty$

Correct answer:

Does not exist

Explanation:

The limit does not exist because the one-sided limits are not equal; $\lim_{x \rightarrow -2^-}f(x) = -\infty$ , whereas $\lim_{x \rightarrow -2^+}f(x) = +\infty$ .

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Example Question #3 : Finding Limits And One Sided Limits

Determine the value of $\lim_{x \to 5^-} \left |-5 \right |$ .

Possible Answers:

$-\infty$

$\infty$

Correct answer:

Explanation:

The expression ${}\lim_{x \to 5^-} \left |-5 \right |$ indicates that all points on the domain are equal to 5 since the absolute value negates negative values.

The $\lim_{x \to 5^-}$ is to search for the limit as the graph approaches x=5 from the left side of the graph. Since the absolute value of negative five is five, the graph approaches five from the left.

The correct answer is .

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Example Question #2 : Finding Limits And One Sided Limits

Evaluate the following limit.

$\small \lim_{x\to 9} \frac{x-9}{\sqrt{x}-3}$

Possible Answers:

Limit does not exist

$\small 6$

$\small 3+\sqrt{3}$

Correct answer:

$\small 6$

Explanation:

This limit can be solved using simple manipulation of the expression inside the limit:

$\\ \small \small \lim_{x\to 9} \frac{x-9}{\sqrt{x}-3}\\ \\=\lim_{x\to 9} \frac{x-9}{\sqrt{x}-3}\cdot \frac{\sqrt{x}+3}{\sqrt{x}+3}\\ \\=\lim_{x\to 9} \frac{x-9}{x-9}\cdot (\sqrt{x}+3)\\ \\=\lim_{x\to 9} \sqrt{x}+3\\ \\=\sqrt{9}+3=6$

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Example Question #3 : Finding Limits And One Sided Limits

Evaluatle the limit:

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

Possible Answers:

DNE

Correct answer:

Explanation:

Consider the domain of the function. Because this equation is a polynomial, x is not restricted by any value. Thus the way to evaluate this limit would simply be to plug the value that x is approaching into the limit equation.

$\lim_{x \to 3} x^2-2x+6=3^2-2*3+6=9-6+6=9$

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Example Question #4 : Finding Limits And One Sided Limits

Evaluate the limit:

$\lim_{x \to -2}\sqrt{x^4+3x+6}$

Possible Answers:

DNE

Correct answer:

Explanation:

$\lim_{x \to -2}\sqrt{x^4+3x+6}=\sqrt{(-2)^4+3(-2)+6}=\sqrt{16-6+6}=\sqrt{16}=4$

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Example Question #5 : Finding Limits And One Sided Limits

Evaluate the limit:

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

Possible Answers:

DNE

$\frac{7}{8}$

$\frac{7}{4}$

Correct answer:

$\frac{7}{8}$

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.

$\lim_{x \to -2}\frac{x^4-2}{2x^2-3x+2}=\frac{(-2)^4-2}{2(-2)^2-3*(-2)+2}=\frac{16-2}{8+6+2}=\frac{14}{16}=\frac{7}{8}$

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Example Question #6 : Finding Limits And One Sided Limits

Evaluate the limit:

$\lim_{x \to 0}\frac{\cos^4 (x)}{5+2x^3}$

Possible Answers:

$\frac{1}{5}$

DNE

$\infty$

Correct answer:

$\frac{1}{5}$

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=0; so we proceed to insert the value of x into the entire equation.

$\lim_{x \to 0}\frac{\cos^4 (x)}{5+2x^3}=\frac{\cos^4 (0)}{5+2(0)^3}=\frac{1}{5}$

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Example Question #7 : Finding Limits And One Sided Limits

Evaluate the limit:

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

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=5; 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.

$\lim_{x \to 5}\frac{x^2-6x+5}{x-5}=\lim_{x \to 5}\frac{(x-5)(x-1)}{x-5}=\lim_{x \to 5}(x-1)=5-1=4$

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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 #1 : Finding Limits And One Sided Limits

Example Question #1 : Finding Limits And One Sided Limits

Example Question #2 : Finding Limits And One Sided Limits

Example Question #3 : Finding Limits And One Sided Limits

Example Question #2 : Finding Limits And One Sided Limits

Example Question #3 : Finding Limits And One Sided Limits

Example Question #4 : Finding Limits And One Sided Limits

Example Question #5 : Finding Limits And One Sided Limits

Example Question #6 : Finding Limits And One Sided Limits

Example Question #7 : Finding Limits And One Sided Limits

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