Student Solutions Manual – Calculus: A
Complete Course 10th Edition Fully Solved
Exercises and Detailed for complex
solutions with verified Answers.
Section A: Limits and Continuity (Questions 1-12)
Question 1
Evaluate: $\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$
A) 0
B) 3
C) 6
D) Does not exist
Answer: C) 6
Rationale: Factor numerator as (x-3)(x+3), cancel (x-3), resulting in $\lim_{x \to 3} (x+3) = 6$ .
Question 2
Find: $\lim_{x \to 0} \frac{\sin 3x}{2x}$
A) 0
B) $\frac{3}{2}$
C) $\frac{2}{3}$
D) Does not exist
Answer: B) $\frac{3}{2}$
Rationale: Using $\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1$, rewrite as $\frac{3}{2}
\cdot \frac{\sin 3x}{3x} = \frac{3}{2} \cdot 1 = \frac{3}{2}$ .
Question 3
Determine $\lim_{x \to \infty} \frac{5x^2 - 3x + 2}{2x^2 + 4x - 7}$
A) 0
B) $\frac{5}{2}$
C) $\infty$
D) $-\infty$
Answer: B) $\frac{5}{2}$
Rationale: Divide numerator and denominator by $x^2$: $\frac{5 - 3/x + 2/x^2}{2 + 4/x -
7/x^2} \to \frac{5}{2}$ as $x \to \infty$.
,Question 4
Evaluate: $\lim_{x \to 0} \frac{\tan 2x}{x}$
A) 0
B) 1
C) 2
D) Does not exist
Answer: C) 2
Rationale: $\lim_{x \to 0} \frac{\tan 2x}{x} = \lim_{x \to 0} \frac{\sin 2x}{x \cos 2x} = \lim_{x
\to 0} \frac{2 \sin 2x}{2x \cos 2x} = 2 \cdot 1 \cdot 1 = 2$.
Question 5
Find $\lim_{x \to 0} \frac{1 - \cos x}{x^2}$
A) 0
B) $\frac{1}{2}$
C) 1
D) Does not exist
Answer: B) $\frac{1}{2}$
Rationale: Using $1 - \cos x = 2\sin^2(x/2)$, we get $\lim_{x \to 0} \frac{2\sin^2(x/2)}{x^2} =
\lim_{x \to 0} \frac{1}{2} \left(\frac{\sin(x/2)}{x/2}\right)^2 = \frac{1}{2} \cdot 1^2 =
\frac{1}{2}$.
Question 6
For what value of k is the following function continuous at x = 2?
𝑥2 − 4
𝑓(𝑥) = { 𝑥 − 2 𝑥 ≠ 2
𝑘 𝑥=2
A) 2
B) 3
C) 4
D) 8
Answer: C) 4
Rationale: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = \lim_{x \to 2} (x+2) = 4$. For continuity, k
must equal this limit.
Question 7
Evaluate: $\lim_{x \to 0^+} x \ln x$
, A) 0
B) $-\infty$
C) 1
D) Does not exist
Answer: A) 0
Rationale: This is a 0·∞ indeterminate form. Rewrite as $\lim_{x \to 0^+} \frac{\ln x}{1/x}$
and apply L'Hôpital's rule: $\lim_{x \to 0^+} \frac{1/x}{-1/x^2} = \lim_{x \to 0^+} (-x) = 0$.
Question 8
Find $\lim_{x \to \infty} \frac{\ln x}{x}$
A) 0
B) 1
C) $\infty$
D) Does not exist
Answer: A) 0
Rationale: Apply L'Hôpital's rule: $\lim_{x \to \infty} \frac{1/x}{1} = \lim_{x \to \infty}
\frac{1}{x} = 0$.
Question 9
Determine $\lim_{x \to 1} \frac{x^3 - 1}{x^2 - 1}$
A) 0
B) $\frac{3}{2}$
C) 1
D) $\frac{2}{3}$
Answer: B) $\frac{3}{2}$
Rationale: Factor: $\frac{(x-1)(x^2+x+1)}{(x-1)(x+1)} = \frac{x^2 + x + 1}{x+1} \to \frac{3}{2}$
as x → 1.
Question 10
Which of the following statements about the function $f(x) = \frac{1}{x-2}$ is true?
A) f is continuous at x = 2
B) f has a removable discontinuity at x = 2
C) f has a non-removable (infinite) discontinuity at x = 2
D) f is continuous for all real numbers
Answer: C) f has a non-removable (infinite) discontinuity at x = 2
Rationale: As x → 2, the denominator approaches 0 while numerator ≠ 0, so |f(x)| → ∞. This
is an infinite discontinuity, which cannot be removed by redefining f at a single point.
Complete Course 10th Edition Fully Solved
Exercises and Detailed for complex
solutions with verified Answers.
Section A: Limits and Continuity (Questions 1-12)
Question 1
Evaluate: $\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$
A) 0
B) 3
C) 6
D) Does not exist
Answer: C) 6
Rationale: Factor numerator as (x-3)(x+3), cancel (x-3), resulting in $\lim_{x \to 3} (x+3) = 6$ .
Question 2
Find: $\lim_{x \to 0} \frac{\sin 3x}{2x}$
A) 0
B) $\frac{3}{2}$
C) $\frac{2}{3}$
D) Does not exist
Answer: B) $\frac{3}{2}$
Rationale: Using $\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1$, rewrite as $\frac{3}{2}
\cdot \frac{\sin 3x}{3x} = \frac{3}{2} \cdot 1 = \frac{3}{2}$ .
Question 3
Determine $\lim_{x \to \infty} \frac{5x^2 - 3x + 2}{2x^2 + 4x - 7}$
A) 0
B) $\frac{5}{2}$
C) $\infty$
D) $-\infty$
Answer: B) $\frac{5}{2}$
Rationale: Divide numerator and denominator by $x^2$: $\frac{5 - 3/x + 2/x^2}{2 + 4/x -
7/x^2} \to \frac{5}{2}$ as $x \to \infty$.
,Question 4
Evaluate: $\lim_{x \to 0} \frac{\tan 2x}{x}$
A) 0
B) 1
C) 2
D) Does not exist
Answer: C) 2
Rationale: $\lim_{x \to 0} \frac{\tan 2x}{x} = \lim_{x \to 0} \frac{\sin 2x}{x \cos 2x} = \lim_{x
\to 0} \frac{2 \sin 2x}{2x \cos 2x} = 2 \cdot 1 \cdot 1 = 2$.
Question 5
Find $\lim_{x \to 0} \frac{1 - \cos x}{x^2}$
A) 0
B) $\frac{1}{2}$
C) 1
D) Does not exist
Answer: B) $\frac{1}{2}$
Rationale: Using $1 - \cos x = 2\sin^2(x/2)$, we get $\lim_{x \to 0} \frac{2\sin^2(x/2)}{x^2} =
\lim_{x \to 0} \frac{1}{2} \left(\frac{\sin(x/2)}{x/2}\right)^2 = \frac{1}{2} \cdot 1^2 =
\frac{1}{2}$.
Question 6
For what value of k is the following function continuous at x = 2?
𝑥2 − 4
𝑓(𝑥) = { 𝑥 − 2 𝑥 ≠ 2
𝑘 𝑥=2
A) 2
B) 3
C) 4
D) 8
Answer: C) 4
Rationale: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = \lim_{x \to 2} (x+2) = 4$. For continuity, k
must equal this limit.
Question 7
Evaluate: $\lim_{x \to 0^+} x \ln x$
, A) 0
B) $-\infty$
C) 1
D) Does not exist
Answer: A) 0
Rationale: This is a 0·∞ indeterminate form. Rewrite as $\lim_{x \to 0^+} \frac{\ln x}{1/x}$
and apply L'Hôpital's rule: $\lim_{x \to 0^+} \frac{1/x}{-1/x^2} = \lim_{x \to 0^+} (-x) = 0$.
Question 8
Find $\lim_{x \to \infty} \frac{\ln x}{x}$
A) 0
B) 1
C) $\infty$
D) Does not exist
Answer: A) 0
Rationale: Apply L'Hôpital's rule: $\lim_{x \to \infty} \frac{1/x}{1} = \lim_{x \to \infty}
\frac{1}{x} = 0$.
Question 9
Determine $\lim_{x \to 1} \frac{x^3 - 1}{x^2 - 1}$
A) 0
B) $\frac{3}{2}$
C) 1
D) $\frac{2}{3}$
Answer: B) $\frac{3}{2}$
Rationale: Factor: $\frac{(x-1)(x^2+x+1)}{(x-1)(x+1)} = \frac{x^2 + x + 1}{x+1} \to \frac{3}{2}$
as x → 1.
Question 10
Which of the following statements about the function $f(x) = \frac{1}{x-2}$ is true?
A) f is continuous at x = 2
B) f has a removable discontinuity at x = 2
C) f has a non-removable (infinite) discontinuity at x = 2
D) f is continuous for all real numbers
Answer: C) f has a non-removable (infinite) discontinuity at x = 2
Rationale: As x → 2, the denominator approaches 0 while numerator ≠ 0, so |f(x)| → ∞. This
is an infinite discontinuity, which cannot be removed by redefining f at a single point.
