Limits and Introduction to Calculus — High School Pre-Calculus practice questions

Evaluate lim_{x→2} (x² − 4)/(x − 2).

1. 4
2. 0
3. Does not exist
4. 2

Answer: A — 4

A) Correct — Factor and cancel: x + 2 → 4. B) Stops at the 0/0 form, which is indeterminate; it means more algebra is needed. C) Treats 0/0 as a final answer; the limit exists once the common factor is cancelled. D) Cancels to x instead of x + 2.

Evaluate lim_{x→0⁺} 1/x.

1. −∞
2. 0
3. ∞
4. 1

Answer: C — ∞

C) Correct — For small positive x, 1/x is large and positive. A) Is the left-hand limit. B) Confuses the input with the output. D) Plugs in x = 1.

For f(x) = (x² − 1)/(x − 1), a table shows f(0.9) = 1.9, f(0.99) = 1.99, f(1.01) = 2.01, f(1.1) = 2.1. Estimate lim_{x→1} f(x).

1. 1
2. Does not exist
3. 0
4. 2

Answer: D — 2

D) Correct — The values approach 2 from both sides. A) Gives the x-value instead of where f is heading. B) f(1) is undefined, but the limit still exists. C) Assumes an undefined value means a limit of 0.

Which limit defines the derivative f'(x)?

1. f(x + h) + f(x)
2. lim_{h→0} [f(x + h) − f(x)]/h
3. [f(b) − f(a)]/(b − a)
4. lim_{x→∞} f(x)

Answer: B — lim_{h→0} [f(x + h) − f(x)]/h

B) Correct — The limit of the difference quotient. A) Adds instead of finding a difference quotient. C) Is an average rate, with no limit. D) Describes end behavior.

A ball's height is s(t) = 16t². Its average velocity on [1, 1 + h] is 32 + 16h. What is its instantaneous velocity at t = 1?

1. 16
2. 48
3. 32 + 16h
4. 32

Answer: D — 32

D) Correct — Let h → 0 in 32 + 16h. A) Uses the coefficient of t². B) Evaluates the speed at t = 1.5. C) Forgets to take the limit.

Evaluate lim_{x→∞} (1 + 1/x)ˣ.

1. 1
2. ∞
3. 0
4. e

Answer: D — e

D) Correct — This limit defines e ≈ 2.718. A) Assumes 1 raised to any power stays 1; here the base is above 1. B) Assumes the power always wins. C) Assumes 1/x vanishing makes everything vanish.

Evaluate lim_{x→0} (sin x)/x.

1. 0
2. 1
3. ∞
4. Does not exist

Answer: B — 1

B) Correct — A standard limit (x in radians). A) Notices sin 0 = 0 but ignores the 0 in the denominator. C) Treats it as division by zero. D) Treats 0/0 as a final answer; the limit exists once the common factor is cancelled.

What does it mean for a function to be continuous at x = a?

f(a) is defined, lim_{x→a} f(x) exists, and the limit equals f(a).

What is the average rate of change of f from x = a to x = b?

[f(b) − f(a)] / (b − a) — slope of the secant line.

What is the limit notation lim_{x→a} f(x) = L?

As x approaches a, f(x) approaches L.