Limit of Trigonometric Functions - Practice Questions

LIMIT OF TRIGONOMETRIC FUNCTIONS — Practice Questions

31 multiple-choice questions on limits of trigonometric functions.
Try to solve each question independently before checking the answer key.

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Instructions:
Choose the best answer. Correct: +4, Wrong: −1, Blank: 0.

Quick Notes — Trigonometric Limits:
\(\displaystyle \lim_{x \to 0} \dfrac{\sin x}{x} = 1\)
\(\displaystyle \lim_{x \to 0} \dfrac{\tan x}{x} = 1\)
\(\displaystyle \lim_{x \to 0} \dfrac{1-\cos x}{x^2} = \dfrac12\)
\(\displaystyle \lim_{x \to 0} \dfrac{\sin mx}{\sin nx} = \dfrac{m}{n}\)
\(\displaystyle \lim_{x \to 0} \dfrac{\sin mx}{\tan nx} = \dfrac{m}{n}\)
For \(x \to a\), use substitution \(p = x-a\) or \(p = x-\frac{\pi}{2}\).

Problem 1

Find \(\displaystyle \lim_{x \to 0} \dfrac{\sin 2x}{\sin 6x} = \cdots \cdot\)

A. \(\dfrac16\) B. \(\dfrac13\) C. \(2\) D. \(3\) E. \(6\)

Problem 2

Find \(\displaystyle \lim_{x \to 0} \dfrac{\sin 7x + \tan 3x-\sin 5x}{\tan 9x-\tan 3x-\sin x} = \cdots \cdot\)

A. \(9\) B. \(7\) C. \(5\) D. \(3\) E. \(1\)

Problem 3

Find \(\displaystyle \lim_{x \to 0} \dfrac{x \cos 2x}{\tan x-\sin 2x} = \cdots \cdot\)

A. \(-1\) B. \(-\dfrac12\) C. \(0\) D. \(\dfrac12\) E. \(1\)

Problem 4

Find \(\displaystyle \lim_{x \to 0} \dfrac{3x + \sin 4x}{5x-\tan 2x} = \cdots \cdot\)

A. \(\dfrac23\) B. \(\dfrac43\) C. \(\dfrac53\) D. \(\dfrac73\) E. \(\dfrac83\)

Problem 5

Find \(\displaystyle \lim_{x \to 0} \dfrac{\sin^3 2x}{\tan^3 \frac12x} = \cdots \cdot\)

A. \(2^3\) B. \(2^4\) C. \(2^5\) D. \(2^6\) E. \(2^7\)

Problem 6

Find \(\displaystyle \lim_{x \to 0} \dfrac{2x^2+x}{\sin x} = \cdots \cdot\)

A. \(-2\) B. \(-1\) C. \(0\) D. \(1\) E. \(2\)

Problem 7

Find \(\displaystyle \lim_{x \to 0} \dfrac{\tan 2x \cdot \tan 3x}{3x^2} = \cdots \cdot\)

A. \(0\) B. \(\dfrac23\) C. \(\dfrac32\) D. \(2\) E. \(6\)

Problem 8

Find \(\displaystyle \lim_{x \to 0} \dfrac{2 \sin^2 \frac12x}{x \tan x} = \cdots \cdot\)

A. \(-2\) B. \(-1\) C. \(-\dfrac12\) D. \(\dfrac12\) E. \(1\)

Problem 9

Find \(\displaystyle \lim_{x \to 0} \dfrac{1-\cos x}{ \sin x} = \cdots \cdot\)

A. \(0\) B. \(\dfrac14\) C. \(\dfrac12\) D. \(1\) E. \(2\)

Problem 10

Find \(\displaystyle \lim_{x \to 0} \dfrac{1-\cos^2 x}{x^2 \cot \left(x-\frac{\pi}{3}\right)} = \cdots \cdot\)

A. \(1\) B. \(0\) C. \(-\dfrac{\sqrt3}{3}\) D. \(-\sqrt2\) E. \(-\sqrt3\)

Problem 11

Find \(\displaystyle \lim_{x \to 0} \dfrac{x \tan 5x}{\cos 2x-\cos 7x} = \cdots \cdot\)

A. \(\dfrac29\) B. \(\dfrac19\) C. \(0\) D. \(-\dfrac19\) E. \(-\dfrac29\)

Problem 12

Find \(\displaystyle \lim_{x \to 0} \dfrac{1-\cos 2x}{x \tan 2x} = \cdots \cdot\)

A. \(-1\) B. \(0\) C. \(1\) D. \(2\) E. \(3\)

Problem 13

Find \(\displaystyle \lim_{x \to 0} \dfrac{1-\cos 4x}{x \sin x} = \cdots \cdot\)

A. \(1\) B. \(2\) C. \(4\) D. \(8\) E. \(16\)

Problem 14

Find \(\displaystyle \lim_{x \to 0} \dfrac{\sin 3x \cdot \tan 5x}{1-\cos 5x} = \cdots \cdot\)

A. \(\dfrac15\) B. \(\dfrac25\) C. \(\dfrac52\) D. \(\dfrac56\) E. \(\dfrac65\)

Problem 15

Find \(\displaystyle \lim_{x \to 0} \left(\dfrac{\sin 4x}{x^2 \tan 2x} - \dfrac{2}{x^2}\right) = \cdots \cdot\)

A. \(-8\) B. \(-4\) C. \(-2\) D. \(2\) E. \(4\)

Problem 16

Find \(\displaystyle \lim_{x \to 0} \dfrac{4x \cos x}{\sin x + \sin 3x} = \cdots \cdot\)

A. \(-1\) B. \(0\) C. \(\dfrac14\) D. \(\dfrac12\) E. \(1\)

Problem 17

Find \(\displaystyle \lim_{x \to \frac{\pi}{4}} \dfrac{\cos 2x}{\sin x-\cos x} = \cdots \cdot\)

A. \(-\sqrt2\) B. \(-\frac12\sqrt2\) C. \(0\) D. \(\frac12\sqrt2\) E. \(\sqrt2\)

Problem 18

Find \(\displaystyle \lim_{x \to 0} \dfrac{\sec 9x-\sec 7x}{\sec 5x-\sec 3x} = \cdots \cdot\)

A. \(0\) B. \(1\) C. \(2\) D. \(3\) E. \(4\)

Problem 19

Find \(\displaystyle \lim_{x \to \frac{\pi}{2}} \dfrac{1-\sin^2 x}{\left(\sin \frac12x-\cos \frac12x\right)^2} = \cdots \cdot\)

A. \(-2\) B. \(-1\) C. \(0\) D. \(1\) E. \(2\)

Problem 20

Find \(\displaystyle \lim_{x \to 3} \dfrac{x \tan (2x-6)}{\sin (x-3)} = \cdots \cdot\)

A. \(1\) B. \(2\) C. \(3\) D. \(6\) E. \(9\)

Problem 21

Find \(\displaystyle \lim_{x \to \frac{\pi}{8}} \dfrac{\sin^2 2x-\cos^2 2x}{\sin 2x-\cos 2x} = \cdots \cdot\)

A. \(-\sqrt2\) B. \(-\frac12\sqrt2\) C. \(0\) D. \(\frac12\sqrt2\) E. \(\sqrt2\)

Problem 22

Find \(\displaystyle \lim_{x \to -2} \dfrac{(x^2-4) \tan (x+2)}{\sin^2 (x+2)} = \cdots \cdot\)

A. \(-4\) B. \(-3\) C. \(0\) D. \(4\) E. \(\infty\)

Problem 23

Find \(\displaystyle \lim_{x \to \frac{\pi}{4}} \dfrac{\left(x-\frac{\pi}{4}\right) \sin \left(3x-\frac{3\pi}{4}\right)}{2(1-\sin 2x)} = \cdots \cdot\)

A. \(\frac32\) B. \(\frac34\) C. \(0\) D. \(-\frac34\) E. \(-\frac32\)

Problem 24

Find \(\displaystyle \lim_{x \to \frac{\pi}{2}} \dfrac{4(x-\pi) \cos^2 x}{\pi(\pi-2x) \tan \left(x-\frac{\pi}{2}\right)} = \cdots \cdot\)

A. \(-2\) B. \(-1\) C. \(0\) D. \(1\) E. \(2\)

Problem 25

Find \(\displaystyle \lim_{x \to 1} \dfrac{(x^2+x-2) \sin (x-1)}{x^2-2x+1} = \cdots \cdot\)

A. \(4\) B. \(3\) C. \(0\) D. \(-\frac14\) E. \(-\frac12\)

Problem 26

Find \(\displaystyle \lim_{x \to 1} \dfrac{(x^2-1) \tan (2x-2)}{\sin^2 (x-1)} = \cdots \cdot\)

A. \(1\) B. \(2\) C. \(4\) D. \(6\) E. \(8\)

Problem 27

Find \(\displaystyle \lim_{x \to 2} \dfrac{(x-2) \cos (\pi x-2\pi)}{\tan (2\pi x-4\pi)} = \cdots \cdot\)

A. \(-\frac{1}{2\pi}\) B. \(-\frac{1}{\pi}\) C. \(0\) D. \(\frac{1}{\pi}\) E. \(\frac{1}{2\pi}\)

Problem 28

Find \(\displaystyle \lim_{x \to 1} \dfrac{\tan (1-x)}{x^3-1} = \cdots \cdot\)

A. \(\frac13\) B. \(-\frac13\) C. \(1\) D. \(-1\) E. \(\frac12\)

Problem 29

Find \(\displaystyle \lim_{x \to \frac{\pi}{2}} \left((\pi-2x) \cdot \tan 5x\right) = \cdots \cdot\)

A. \(\frac45\) B. \(\frac35\) C. \(\frac25\) D. \(\frac12\) E. \(\frac14\)

Problem 30

Find \(\displaystyle \lim_{x \to 0} \dfrac{x^3}{\sqrt{1+\sin x}-\sqrt{1+\tan x}} = \cdots \cdot\)

A. \(-4\) B. \(-2\) C. \(0\) D. \(2\) E. \(4\)

Problem 31

Find \(\displaystyle \lim_{x \to \pi} \dfrac{\sqrt{5+\cos x}-2}{(\pi-x)^2} = \cdots \cdot\)

A. \(\frac{1}{10}\) B. \(\frac18\) C. \(\frac13\) D. \(\frac12\) E. \(1\)