# Mathematics Questions and Answers – Limits of Trigonometric Functions

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This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Limits of Trigonometric Functions”.

1. What is the value of $$\lim\limits_{y \rightarrow \pi/2}\frac{sin⁡ x}{x}$$?
a) $$\frac{2}{\pi}$$
b) $$\frac{\pi}{2}$$
c) 1
d) 0

Explanation: sin ⁡$$\frac{\pi}{2}$$ = 1
$$\lim\limits_{y \rightarrow \pi/2}\frac{sin⁡x}{x} = \frac{sin⁡\frac{π}{2}}{\frac{\pi}{2}}$$
= $$\frac{1}{\frac{\pi}{2}}$$
= $$\frac{2}{\pi}$$

2. What is the value of $$\lim\limits_{y \rightarrow 0}\frac{sin⁡3y}{3y}$$?
a) 0
b) 1
c) 3
d) $$\frac{1}{3}$$

Explanation: We know that $$\lim\limits_{x \rightarrow 0}\frac{sin⁡x}{x}$$ = 1.
Here x tends to 3y.
Also, since this is of the form $$\frac{0}{0}$$, we use L’Hospital’s rule and differentiate the numerator and denominator separately.
= $$\lim\limits_{y \rightarrow 0}\frac{3\, cos\, 3y}{3}$$
= 1

3. What is the value of $$\lim\limits_{x \rightarrow 0}\frac{x^2sec x}{sin⁡ x}$$?
a) 3
b) 2
c) 1
d) 0

Explanation: $$\lim\limits_{x \rightarrow 0}\frac{x}{sin⁡ x}$$x $$\lim\limits_{x \rightarrow 0}⁡\frac{x}{cos⁡ x}$$
= 1 x 0
= 0
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4. What is the value of $$\lim\limits_{x \rightarrow 0}\frac{x \,tanx}{cot\, x}$$?
a) 0
b) 1
c) 2
d) $$\frac{1}{2}$$

Explanation: $$\lim\limits_{x \rightarrow 0}\frac{x tanx}{cot x}$$ = $$\lim\limits_{x \rightarrow 0}\frac{x\frac{sin⁡ x}{cos ⁡x}}{\frac{cos⁡ x}{sin⁡ x}}$$
= $$\lim\limits_{x \rightarrow 0}$$ ⁡x
= 0

5. What is the value of $$\lim\limits_{x \rightarrow \infty}\frac{x sin⁡\frac{2}{x}}{2}$$?
a) 1
b) 2
c) $$\frac{1}{2}$$
d) Limit does not exist

Explanation:
This is of the form $$\frac{0}{0}$$, so we use L’Hospital’s rule.
= $$\lim\limits_{x \rightarrow \infty}\frac{\frac{-2}{x^2}cos⁡\frac{2}{x}}{\frac{-2}{x^2}}$$
= $$\lim\limits_{x \rightarrow \infty}$$cos$$\frac{2}{x}$$
= 1

6. Which of the following limits does not yield 1?
a) $$\lim\limits_{x \rightarrow 0}\frac{⁡sin x}{x}$$
b) $$\lim\limits_{x \rightarrow 0}\frac{⁡tan x}{cot x}$$
c) $$\lim\limits_{x \rightarrow 0}(\frac{1}{e^x}+cos⁡ x)$$
d) $$\lim\limits_{x \rightarrow 0}$$ x cosec x

Explanation: $$\lim\limits_{x \rightarrow 0}(\frac{1}{e^x} + sin⁡ x) = \frac{1}{e^0}$$ + cos (0)
= 1 + 1
= 2

7. What is the value of $$\lim\limits_{y \rightarrow 0}$$(32 x2 cosec2 ⁡4x)?
a) 1
b) 4
c) 2
d) 3

Explanation: The limit can be written as, $$\lim\limits_{x \rightarrow 0}\frac{32x^2}{sin^2⁡4x}$$
= 2 x $$\lim\limits_{x \rightarrow 0}\frac{4x}{sin 4x}$$ x $$\lim\limits_{x \rightarrow 0}\frac{4x}{sin 4x}$$
= 2 x 1 x 1
= 2

8. What is the value of the limit f(x) = $$\frac{sin^2⁡x+\sqrt 2 sin ⁡x}{x^2-4x}$$ if x approaches 0?
a) $$\frac{1}{\sqrt 2}$$
b) $$\frac{-1}{\sqrt 2}$$
c) $$\frac{-1}{2\sqrt 2}$$
d) $$\frac{1}{2\sqrt 2}$$

Explanation: This is of the form $$\frac{0}{0}$$, therefore we use L’Hospital’s rule and differentiate the numerator and denominator.
= $$\lim\limits_{x \rightarrow 0}\frac{2sin⁡ \,x cos \,⁡x + cos \,⁡x \sqrt 2}{2x – 4}$$
= $$\frac{0+\sqrt 2}{-4}$$
= $$\frac{-1}{2\sqrt 2}$$

9. What is the value of the $$\lim\limits_{x \rightarrow \frac{3\pi}{2}}\frac{cos⁡ x sin⁡ x}{sin⁡2x}$$?
a) $$\frac{-1}{2}$$
b) $$\frac{1}{2}$$
c) $$\frac{1}{4}$$
d) $$\frac{-1}{4}$$

Explanation: $$\lim\limits_{x \rightarrow \frac{3\pi}{2}}\frac{cos⁡ x sin⁡ x}{sin⁡2x}$$ =$$\lim\limits_{x \rightarrow \frac{3\pi}{2}}\frac{cos⁡ x sin⁡ x}{2 cos x sin⁡ x}$$
= $$\frac{1}{2}$$

10. What is the value of the limit $$\lim\limits_{x \rightarrow \frac{\pi}{2}}\frac{sin^2⁡x-1}{cos ⁡x}$$?
a) 0
b) 4
c) 1
d) Limit does not exist

Explanation: $$\lim\limits_{x \rightarrow \frac{\pi}{2}}\frac{sin^2⁡x-1}{cos ⁡x}$$ = $$\lim\limits_{x \rightarrow \frac{\pi}{2}}\frac{-cos^2 x}{cos ⁡x}$$
=$$\lim\limits_{x \rightarrow \frac{\pi}{2}}$$ -cosx
= 0

11. What is the value of the limit $$\lim\limits_{x \rightarrow 0}\frac{sin^2⁡x}{x^2}$$?
a) 2
b) 1
c) Limit does not exist
d) 4

Explanation: $$\lim\limits_{x \rightarrow 0}\frac{sin^2⁡x}{x^2}$$ =
= ($$\lim\limits_{x \rightarrow 0}\frac{sin ⁡x}{x}$$ x $$\lim\limits_{x \rightarrow 0}\frac{sinx}{x}$$)
We apply L’Hospital’s rule and differentiate numerator and denominator.
= ($$\lim\limits_{x \rightarrow 0}\frac{cos x}{1}$$ x $$\lim\limits_{x \rightarrow 0}\frac{cos x}{1}$$)
= 1

12. What is the value of $$\lim\limits_{x \rightarrow 0}\frac{e^x(sin^2⁡ x)}{x^3}$$?
a) 2
b) 3
c) 1
d) 0

Explanation: $$\lim\limits_{x \rightarrow 0}\frac{sin^2⁡ x}{x^2}$$ x $$\lim\limits_{x \rightarrow 0}\frac{e^x}{x}$$
We apply L’Hospital’s rule and differentiate numerator and denominator.
= 1 x $$\lim\limits_{x \rightarrow 0}\frac{e^x}{1}$$
= 1

Sanfoundry Global Education & Learning Series – Mathematics – Class 11.

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