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Calculus 2 : Finding Limits and One-Sided Limits

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Example Questions

Example Question #231 : Finding Limits And One Sided Limits

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 #232 : Finding Limits And One Sided Limits

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

Possible Answers:

Does not exist

$-\infty$

$\infty$

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 #233 : Finding Limits And One Sided Limits

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

Possible Answers:

Does not exist

$-\infty$

$\infty$

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 #234 : Finding Limits And One Sided Limits

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 #235 : Finding Limits And One Sided Limits

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 #236 : Finding Limits And One Sided Limits

Graph 20150825 151852

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

Possible Answers:

$\infty$

Does not exist

$-\infty$

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 #279 : Limits

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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Example Question #236 : Finding Limits And One Sided Limits

Considering the following piecewise function, what is $\lim_{x \rightarrow 11^+} y$ , $y = \begin{Bmatrix}8 & x < 11 \\ 2 & x = 11 \\-6 & x \geq 11\end{Bmatrix}$ .

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=11 , from the right. The limit exists, and it corresponds to the function . Since this function has no inputs, the limit is .

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Example Question #237 : Finding Limits And One Sided Limits

What is the $\lim_{x \rightarrow 0^-} y$ , for $y=\frac{x^2+2}{x-2}$ .

Possible Answers:

Does not exist

Correct answer:

Explanation:

As you go to x=0 from the left, what value do you get closer to?

The limit of x=0 from the left is

$\frac{0^2 + 2}{0-2} = -1$ .

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Example Question #238 : Finding Limits And One Sided Limits

Evaluate the following limit:

$\lim_{x\rightarrow \pi^{-}}f(x), f(x)=\left\{\begin{matrix} \cos(x), x\leq \pi\\ x, x> \pi \end{matrix}\right.$

Possible Answers:

$-\pi$

$\pi$

The limit does not exist

Correct answer:

Explanation:

To evaluate the limit, we must first see whether the limit is right or left sided. The negative sign "exponent" on $\pi$ indicates that we are evaluating the limit from the left side, using values slightly less than $\pi$ . So, the part of the piecewise function we will use is the first one; when we evaluate the limit, we get an answer of

( $\cos(\pi)=-1$ ).

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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 #231 : Finding Limits And One Sided Limits

Example Question #232 : Finding Limits And One Sided Limits

Example Question #233 : Finding Limits And One Sided Limits

Example Question #234 : Finding Limits And One Sided Limits

Example Question #235 : Finding Limits And One Sided Limits

Example Question #236 : Finding Limits And One Sided Limits

Example Question #279 : Limits

Example Question #236 : Finding Limits And One Sided Limits

Example Question #237 : Finding Limits And One Sided Limits

Example Question #238 : Finding Limits And One Sided Limits

All Calculus 2 Resources

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