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AP Calculus BC Unit 1: Limits and Continuity

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Unit 1: Limits and Continuity

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Question 1

The function $f (x) = \frac{x ^{2} - 4}{x - 2}$ has what type of discontinuity at $x = 2$ ?

A.Vertical asymptoteB.Jump discontinuityC.Removable discontinuityD.Essential discontinuity

Question 2

Find all horizontal asymptotes of $f (x) = \frac{3 x ^{2} - 1}{4 x ^{4} + 5 x ^{2}}$ .

y = \frac{3}{2}

onlyB.

y = \frac{3}{4}

y = \frac{3}{2}

and

y = - \frac{3}{2}

y = \frac{3}{2}

Question 3

For the function $f (x) = \frac{∣ x ^{2} - 4∣}{x ^{2} - 4}$ , evaluate $lim_{x \to 2^{-}} f (x)$ and $lim_{x \to 2^{+}} f (x)$ , then classify the discontinuity at $x = 2$ .

A.Both limits equal

1

; removable discontinuity.B.

lim_{x \to 2^{-}} f (x) = - 1

lim_{x \to 2^{+}} f (x) = 1

; jump discontinuity.C.The limit does not exist and

f

has an infinite discontinuity.D.

lim_{x \to 2^{-}} f (x) = 1

lim_{x \to 2^{+}} f (x) = - 1

; jump discontinuity.

Question 4

For what value of $c$ is $f (x) = {x^{2} + c 3 x - 1 x < 2 x \geq 2$ continuous at $x = 2$ ?

c = 1

c = - 1

c = 3

c = 5

Question 5

Evaluate $x \to \infty lim (1 + \frac{3}{x})^{x}$ .

1

e^{3}

3 e

\infty

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Question 6

For $p (x) = \frac{sin x}{x ( x - π )}$ , identify and classify all discontinuities on $(- \infty, \infty)$ .

A.Vertical asymptote at

x = 0

; vertical asymptote at

x = π

.B.Removable at

x = 0

; removable at

x = π

.C.Removable at

x = 0

; vertical asymptote at

x = π

.D.Jump discontinuity at

x = 0

; removable at

x = π

Question 7

Evaluate $lim_{x \to 0} \frac{1 - cos x}{x ^{2}}$ .

0

\frac{1}{2}

1

2

Question 8

Evaluate $lim_{x \to \infty} \frac{4 x ^{3} - 2 x + 7}{6 x ^{3} + 5 x ^{2} - 1}$ .

0

\frac{4}{5}

\frac{2}{3}

\infty

Question 9

The Squeeze Theorem guarantees $lim_{x \to 0} x^{2} sin (\frac{1}{x}) = 0$ because:

sin (1/ x) \to 1

x \to 0

, so the product has limit

0 \cdot 1 = 0

.B.

- x^{2} \leq x^{2} sin (1/ x) \leq x^{2}

and both bounds approach

0

.C.L'Hôpital's Rule applies since the expression has

0 \cdot \infty

form.D.

x^{2} sin (1/ x) = 0

for all

x \neq = 0

Question 10

Evaluate $lim_{x \to 0} \frac{sin ( 3 x )}{5 x}$ .

\frac{1}{5}

\frac{3}{5}

\frac{5}{3}

0

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Question 11

Determine $lim_{x \to 0^{+}} \frac{ln x}{cot x}$ .

- \infty

0

1

+ \infty

Question 12

Let $f$ be continuous on $R$ with $f (x) = \frac{sin ( x ^{2} )}{x}$ for $x \neq = 0$ . What must $f (0)$ equal?

0

1

- 1

f (0)

can be any real number.

AP Calculus BC Unit 1: Limits and Continuity — Free with Answer Explanations | Test Practice Hub