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Calculus 2 : Limits

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Example Questions

Example Question #271 : Calculus Ii

Considering the following piecewise function, what is $\lim_{x \rightarrow 2^+} y,$ $y= \begin{Bmatrix}3 & x < 2 \\ -2 & x \geq 2\end{Bmatrix}$

Possible Answers:

Does not exist

Correct answer:

Explanation:

In general, when you are looking for $\lim_{x \rightarrow a^+} y$ you are looking to see whether the limit of y exists to the right, and if it does, what is the value.

Solution:

In this case, we want to see the limit at x=2 , from the right. The limit exists, and the value is

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Example Question #272 : Calculus Ii

Considering the following piecewise function, what is $\lim_{x \rightarrow 6^-} y$ ,

$y = \begin{Bmatrix}x+2 & x < 6 \\ x^2 & x \geq 6\end{Bmatrix}$

Possible Answers:

Does not exist

Correct answer:

Explanation:

In general, when you are looking fo r $\lim_{x \rightarrow a^-} y$ you are looking to see whether the limit of y exists to the left, and if it does, what is the value.

Solution:

In this case, we want to see the limit at x=6 , from the left. The limit exists, as reflected by the function x+2

6+2 = 8

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Example Question #273 : Calculus Ii

Considering the following piecewise function, what is $\lim_{x \rightarrow 5^-} y$ , $y = \begin{Bmatrix}x+2 & x < 5 \\ x^2 + 4 & x = 5 \\x^2 + x & x \geq 5\end{Bmatrix}$

Possible Answers:

Does not exist

Correct answer:

Explanation:

In general, when you are looking for $\lim_{x \rightarrow a^-}$ y you are looking to see whether the limit of y exists to the left, and if it does, what is the value.

Solution:

In this case, we want to see the limit at x=5 , from the left. The limit exists and corresponds to the function x + 2 .

Plug in 5 for x and solve to get the value of the limit from the left.

5+2 = 7

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Example Question #274 : Calculus Ii

What is the $\lim_{x \rightarrow 2^-} y$ , for y=x^3

Possible Answers:

Does not exist

Correct answer:

Explanation:

As you go to x=2 from the left, what value do you get closer to? The limit of x=2 from the left for the function x^3 is .

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Example Question #275 : Calculus Ii

What is the $\lim_{x \rightarrow 5^-} y$ , for $y=\frac{x+5}{x-5}$

Possible Answers:

$\infty$

$-\infty$

Does not exist

Correct answer:

$-\infty$

Explanation:

As you go to x=5 , the function will have a 0 for the denominator, which is not allowed.

As we go to 5 from the left, the function is tending towards negative infinity. It is a negative because when going to 5 from the left, the function will always be less than 5, hence negative.

So $\lim_{x \rightarrow 5^-} y = -\infty$

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Example Question #276 : Calculus Ii

What is the $\lim_{x \rightarrow 6^+} y$ , for $y=\frac{x^2+x}{x-6}$

Possible Answers:

$-\infty$

$\infty$

Does not exist

Correct answer:

$\infty$

Explanation:

As you go to x=6, the function will have a 0 for the denominator.

As we go to 6 from the right, the function is tending towards infinity. It is positive because x wil always be greater than 6, since we're going from the right, and the function goes to infinity because the denominator is getting smaller and smaller.

So, $\lim_{x \rightarrow 6^+} y = \infty$

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Example Question #277 : Calculus Ii

Graph 20150825 144550

What is the $\lim_{x \rightarrow 9^-}$ y, for $y=\frac{x^3+8}{x-9}$

Possible Answers:

$\infty$

Does not exist

$-\infty$

Correct answer:

$-\infty$

Explanation:

As shown by the graph, the limit as x goes to 9 from the left is negative infinity. It is negative because x is aways less than 9.

Answer: $-\infty$

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Example Question #278 : Calculus Ii

Graph 20150825 150219

What is the $\lim_{x \rightarrow 11^-} y$ , for $y=\frac{x^2+4}{x-11}$

Possible Answers:

$-\infty$

$\infty$

Does not exist

Correct answer:

$-\infty$

Explanation:

As shown by the graph, the limit as x goes to 11 from the left is negative infinity. It is negative because x is aways less than 11. To make sure, we can see that the function has denominator x-11 , which becomes smaller and smaller as x goes to 11 from the left.

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Example Question #279 : Calculus Ii

Graph 20150825 151852

What is the $\lim_{x \rightarrow \frac{1}{5}^+} y$ , for $y=\frac{x^2+2x}{5x-1}$

Possible Answers:

$-\infty$

$\infty$

Does not exist

Correct answer:

$\infty$

Explanation:

As x goes to $\frac{1}{5}$ from the right, the function goes to infinity. We know that $\frac{1}{5}$ is the value that brings the limit to infinity because the denominator of the function is 5x-1 , and we want the value of x that turns the denominator into 0.

Answer: the limit is $\infty$

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Example Question #280 : Calculus Ii

Graph 20150825 152620

What is the $\lim_{x \rightarrow -\frac{1}{2}^-} y$ , for $y=\frac{x^2+6x}{4x^2-1}$

Possible Answers:

$-\infty$

$\infty$

Does not exist

Correct answer:

$-\infty$

Explanation:

When $x = -\frac{1}{2}^-$ , the graph tends to $-\infty$ .

To confirm this, we can look at the equation of the function and see when the denominator is 0. The denominator, 4x^2-1 , is 0 when $x = -\frac{1}{2}$ and $x = \frac{1}{2}$ .

We also know that y goes to negative infinity when going to the left because as x tends infinitely closer to $-\frac{1}{2}$ , the absolute value of the function goes to infinity.

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Calculus 2 : Limits

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #271 : Calculus Ii

Example Question #272 : Calculus Ii

Example Question #273 : Calculus Ii

Example Question #274 : Calculus Ii

Example Question #275 : Calculus Ii

Example Question #276 : Calculus Ii

Example Question #277 : Calculus Ii

Example Question #278 : Calculus Ii

Example Question #279 : Calculus Ii

Example Question #280 : Calculus Ii

All Calculus 2 Resources

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