Question 1

Suppose the position of a particle is given by $$x(t) = 16t^2$. What is the average velocity of the particle over the interval $$[2, 2.01]$$?

Accepted Answer

$$64.32$$

Question 2

Given $$x(t) = 16t^2$, what is the instantaneous velocity at t = 2$?

Accepted Answer

$$64$$

Question 3

Which of the following best describes the meaning of $$\lim_{x 	o a} f(x) = L$$?

Accepted Answer

As $$x$$ approaches $$a$$, $$f(x)$$ becomes arbitrarily close to $$L$$

Question 4

Estimate $$\lim_{x 	o 2} \frac{x^2$ - 2x - 3}{x - 2}$$ using a table of values. What is the limit?

Accepted Answer

$$4$$

Question 5

Calculate $$\lim_{x 	o 2} \frac{x^2$ - 2x - 3}{x - 2}$$ algebraically.

Accepted Answer

$$4$$

Question 6

For the function $$x(t) = 16t^2$, what is the average velocity over $$[2, 2+h]$$ in terms of h$?

Accepted Answer

$$16h + 64$$

Question 7

What does $$\lim_{h 	o 0} (16h + 64)$$ represent in the context of velocity for $$x(t) = 16t^2$ at t = 2$?

Accepted Answer

The instantaneous velocity at $$t = 2$$

Question 8

Given $$f(x) = \frac{x^2$ - 2x - 3}{x - 2}$$, which of the following is true about $$f(x) at$$ $$x = 2$$?

Accepted Answer

$$f(2) is $$undefined

Question 9

Which statement is true about the left-sided limit $$\lim_{x 	o a^-} f(x)$$?

Accepted Answer

It is the value $$f(x)$$ approaches as $$x$$ approaches $$a$$ from the left

Question 10

Suppose a car's position is given by $$x(t) = 16t^2$. If the car travels from t = 2$ to t = 2.001$, what is the best estimate for its average velocity over this interval?

Accepted Answer

$$64.016$$

Question 11

If $$\lim_{x 	o 3} f(x) = 1$$ and $$\lim_{x 	o 3^-} f(x) = -1$$, what can you conclude about $$f(x) at$$ $$x = 3$$?

Accepted Answer

The two-sided limit does not exist at $$x = 3$$

Question 12

Which of the following is a correct interpretation of the right-sided limit $$\lim_{x 	o a^+} f(x)$$?

Accepted Answer

The value $$f(x)$$ approaches as $$x$$ approaches $$a$$ from the right

Question 13

A graph of $$f(x)$$ shows a jump at $$x = 1. $$What is $$\lim_{x 	o 1} f(x)$$?

Accepted Answer

Does not exist

Question 14

If $$\lim_{x 	o 0} f(x) = 3$$, which of the following statements is true?

Accepted Answer

As $$x$$ gets arbitrarily close to $$0$$, $$f(x)$$ gets arbitrarily close to $$3$$

Question 15

Suppose $$f(x) is $$defined everywhere except at $$x = 2$$, and $$\lim_{x 	o 2} f(x) = 4. $$Which of the following is true?

Accepted Answer

The limit exists even though $$f(2) is $$undefined

Question 16

Given $$x(t) = 16t^2$, what is the instantaneous velocity at t = a$?

Accepted Answer

$$32t$$

Question 17

Which of the following best describes the process of estimating a limit using a table of values?

Accepted Answer

Calculate $$f(x)$$ for values of $$x$$ increasingly close to the point of interest

Question 18

If $$\lim_{x 	o 1^-} f(x) = 2$$ and $$\lim_{x 	o 1^+} f(x) = 3$$, what is $$\lim_{x 	o 1} f(x)$$?

Accepted Answer

Does not exist

Question 19

A patient’s heart rate is measured every second and modeled by $$x(t) = 16t^2$. What is the best estimate for the instantaneous rate of change at t = 2$?

Accepted Answer

$$64$$

Introduction to Limits in Calculus: Concepts, Examples, and Applications

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