Q: Which of the following statements about limits is TRUE?

A limit as $$x \to a$$ exists if and only if the left-hand and right-hand limits are equal.

Q: If $$\lim_{x \to 3^-} f(x) = 2$$ and $$\lim_{x \to 3^+} f(x) = 4$$, what can you conclude about $$\lim_{x \to 3} f(x)$$?

$$\lim_{x \to 3} f(x)$$ does not exist

Question 1

What is the value of $$\lim_{x 	o 2} (x^2$ + x - 2)$$?

Accepted Answer

$$4$$

Question 2

Which of the following statements about limits is TRUE?

Accepted Answer

A limit as $$x 	o a$$ exists if and only if the left-hand and right-hand limits are equal.

Question 3

Evaluate $$\lim_{y 	o 2} \frac{y^2$ + 5y + 6}{y + 2}.$$

Accepted Answer

$$5$$

Question 4

A flea jumps half the remaining distance to a dog with each jump. If the initial distance is 1 meter, what is the limit of the flea's distance from the dog after infinitely many jumps?

Accepted Answer

$$0$$ meters

Question 5

Given $$f(x) = \frac{x^2$ - x - 2}{x - 2}$$, what is $$\lim_{x 	o 2} f(x)$$?

Accepted Answer

$$3$$

Question 6

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

Accepted Answer

The value of $$f(x) as$$ $$x$$ approaches $$a$$ from the left.

Question 7

If $$\lim_{x 	o 3^-} f(x) = 2$$ and $$\lim_{x 	o 3^+} f(x) = 4$$, what can you conclude about $$\lim_{x 	o 3} f(x)$$?

Accepted Answer

$$\lim_{x 	o 3} f(x)$$ does not exist

Question 8

Evaluate $$\lim_{x 	o 0} \frac{\sin 2x}{3x - x^2$}.$$

Accepted Answer

$$\frac{2}{3}$$

Question 9

Suppose $$\lim_{x 	o 2} f(x) = -2$$ and $$\lim_{x 	o 2} g(x) = 3. $$What is $$\lim_{x 	o 2} [f(x) + g(x)]$$?

Accepted Answer

$$1$$

Question 10

A chemical reaction's rate of change of concentration $$C$$ with respect to time $$t is $$modeled by $$C(t) = 5t^2$ - 3t + 2. $$What is the instantaneous rate of change at $$t = 1$$?

Accepted Answer

$$7$$

Question 11

Which of the following is NOT a valid method for evaluating a limit?

Accepted Answer

Ignoring discontinuities

Question 12

If $$\lim_{x 	o a} f(x)$$ exists and $$f(a) is $$not defined, what can you conclude?

Accepted Answer

The limit exists but the function is not defined at $$x = a.$$

Question 13

Evaluate $$\lim_{x 	o 1} \frac{x^3$ - 8}{x^2$ + x - 6}.$$

Accepted Answer

$$-7$$

Question 14

A patient’s blood glucose level $$G(t) (in mg/dL) is $$modeled by $$G(t) = 100 - 10t$$ for $$t in $$hours after a meal. What is the average rate of change in glucose level from $$t = 1 to$$ $$t = 3$$?

Accepted Answer

$$-5 mg/dL $$per hour

Question 15

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

Accepted Answer

The value $$f(x)$$ approaches as $$x$$ gets arbitrarily close to $$a.$$

Question 16

If $$\lim_{x 	o 0} \frac{\sin x}{x} = 1$$, what is $$\lim_{x 	o 0} \frac{\sin 3x}{x}$$?

Accepted Answer

$$3$$

Question 17

A function $$f(x) is $$defined as $$f(x) = \begin{cases} 2x+1 & x \u003c 3 \ x^2$ & x \geq 3 \end{cases}. $$What is $$\lim_{x 	o 3^-} f(x)$$?

Accepted Answer

$$7$$

Question 18

Suppose $$\lim_{x 	o 2} f(x) = 5$$ and $$\lim_{x 	o 2} g(x) = -1. $$What is $$\lim_{x 	o 2} [f(x) \cdot g(x)]$$?

Accepted Answer

$$-5$$

Question 19

Which of the following real-world scenarios best illustrates the concept of a limit?

Accepted Answer

A ball bouncing and losing half its height with each bounce, approaching the ground.

Lesson 2.1: Rates of Change and Limits – Calculus Study Notes

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