Calculus I · Limits and Continuity · practice

Trigonometric Limit Practice Problems

Visual study stop

Read the picture before the symbols

Pause at each graph long enough to say what the input is doing, what the output is doing, and which feature supports the next mathematical decision.

Graph of sin(x) divided by x with an open point at (0, 1). — **Anchor every rewrite to the fundamental shape**
The open point marks an undefined quotient at zero while the graph approaches one from both sides. In each exercise, rewrite until the same angle appears in the sine and its denominator.

Section 3 Exercises

A. Squeeze Theorem

Exercise 1

If $-x^4\le f(x)\le x^4$ near zero, find $\lim_{x\to0}f(x)$ .

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$0$ .

Answer 1 from the source-traced unit appendix.

Exercise 2

If $2-x^2\le g(x)\le2+x^2$ near zero, find the limit.

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$2$ .

Answer 2 from the source-traced unit appendix.

Exercise 3

Evaluate $\lim_{x\to0}x^3\sin(1/x)$ .

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$0$ .

Answer 3 from the source-traced unit appendix.

Exercise 4

Evaluate $\lim_{x\to0}x^2\cos(7/x)$ .

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$0$ .

Answer 4 from the source-traced unit appendix.

Exercise 5

Evaluate $\lim_{x\to0}|x|\sin(4/x)$ .

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$0$ .

Answer 5 from the source-traced unit appendix.

Exercise 6

Evaluate $\lim_{x\to0}x\cos(1/x^2)$ .

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$0$ .

Answer 6 from the source-traced unit appendix.

Exercise 7

Suppose $|f(x)|\le5|x-2|$ . Find $\lim_{x\to2}f(x)$ .

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$0$ .

Answer 7 from the source-traced unit appendix.

Exercise 8

Explain why boundedness of $f$ alone is not enough to conclude $\lim_{x\to0}f(x)=0$ .

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A bounded function need not approach zero; for example, $\sin(1/x)$ remains bounded but has no limit.

Answer 8 from the source-traced unit appendix.

B. Fundamental sine limits

Exercise 9

$\displaystyle\lim_{x\to0}\frac{\sin x}{x}$

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$1$ .

Answer 9 from the source-traced unit appendix.

Exercise 10

$\displaystyle\lim_{x\to0}\frac{\sin(2x)}{x}$

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$2$ .

Answer 10 from the source-traced unit appendix.

Exercise 11

$\displaystyle\lim_{x\to0}\frac{\sin(7x)}{3x}$

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$7/3$ .

Answer 11 from the source-traced unit appendix.

Exercise 12

$\displaystyle\lim_{x\to0}\frac{\sin(5x)}{\sin(2x)}$

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$5/2$ .

Answer 12 from the source-traced unit appendix.

Exercise 13

$\displaystyle\lim_{x\to0}\frac{\sin(3x)}{\sin(8x)}$

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$3/8$ .

Answer 13 from the source-traced unit appendix.

Exercise 14

$\displaystyle\lim_{t\to0}\frac{\sin(\pi t)}{t}$

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$\pi$ .

Answer 14 from the source-traced unit appendix.

Exercise 15

$\displaystyle\lim_{x\to0}\frac{x}{\sin(4x)}$

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$1/4$ .

Answer 15 from the source-traced unit appendix.

Exercise 16

$\displaystyle\lim_{x\to0}\frac{3x}{\sin(5x)}$

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$3/5$ .

Answer 16 from the source-traced unit appendix.

C. Tangent and cosine limits

Exercise 17

$\displaystyle\lim_{x\to0}\frac{\tan x}{x}$

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$1$ .

Answer 17 from the source-traced unit appendix.

Exercise 18

$\displaystyle\lim_{x\to0}\frac{\tan(4x)}{x}$

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$4$ .

Answer 18 from the source-traced unit appendix.

Exercise 19

$\displaystyle\lim_{x\to0}\frac{\tan(3x)}{2x}$

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$3/2$ .

Answer 19 from the source-traced unit appendix.

Exercise 20

$\displaystyle\lim_{x\to0}\frac{1-\cos x}{x}$

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$0$ .

Answer 20 from the source-traced unit appendix.

Exercise 21

$\displaystyle\lim_{x\to0}\frac{1-\cos x}{x^2}$

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$1/2$ .

Answer 21 from the source-traced unit appendix.

Exercise 22

$\displaystyle\lim_{x\to0}\frac{1-\cos(4x)}{x^2}$

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$8$ .

Answer 22 from the source-traced unit appendix.

Exercise 23

$\displaystyle\lim_{x\to0}\frac{\sin^2x}{x^2}$

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$1$ .

Answer 23 from the source-traced unit appendix.

Exercise 24

$\displaystyle\lim_{x\to0}\frac{\sin^2(3x)}{x^2}$

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$9$ .

Answer 24 from the source-traced unit appendix.

D. Mixed trigonometric limits

Exercise 25

$\displaystyle\lim_{x\to0}\frac{\sin(2x)\sin(3x)}{x^2}$

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$6$ .

Answer 25 from the source-traced unit appendix.

Exercise 26

$\displaystyle\lim_{x\to0}\frac{\tan(5x)}{\sin(2x)}$

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$5/2$ .

Answer 26 from the source-traced unit appendix.

Exercise 27

$\displaystyle\lim_{x\to0}\frac{1-\cos(2x)}{\sin^2x}$

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$2$ .

Answer 27 from the source-traced unit appendix.

Exercise 28

$\displaystyle\lim_{x\to0}\frac{\sin(4x)}{x\cos x}$

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$4$ .

Answer 28 from the source-traced unit appendix.

Exercise 29

$\displaystyle\lim_{x\to0}\frac{\sin x}{x(1+\cos x)}$

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$1/2$ .

Answer 29 from the source-traced unit appendix.

Exercise 30

$\displaystyle\lim_{x\to0}\frac{\tan x-\sin x}{x}$

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$0$ .

Answer 30 from the source-traced unit appendix.

Exercise 31

$\displaystyle\lim_{x\to0}\frac{1-\cos x}{x\sin x}$

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$1/2$ .

Answer 31 from the source-traced unit appendix.

Exercise 32

$\displaystyle\lim_{x\to0}\frac{\sin(3x)}{\tan(5x)}$

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$3/5$ .

Answer 32 from the source-traced unit appendix.

E. Reasoning and exam practice

Exercise 33

Explain why radians are required for $\lim_{x\to0}\sin x/x=1$ .

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Radian measure equals unit-circle arc length; the geometric squeeze produces ratio $1$ only in radians.

Answer 33 from the source-traced unit appendix.

Exercise 34

Derive $\lim_{x\to0}\tan x/x=1$ from the sine limit.

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$\tan x/x=(\sin x/x)(1/\cos x)\to1$ .

Answer 34 from the source-traced unit appendix.

Exercise 35

Derive $\lim_{x\to0}(1-\cos x)/x^2=1/2$ .

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Rationalize and use $(\sin x/x)^2/(1+\cos x)\to1/2$ .

Answer 35 from the source-traced unit appendix.

Exercise 36

A student writes $\sin(5x)=5x$ . Explain what is wrong and how to use the limit correctly.

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$\sin(5x)$ is not equal to $5x$ ; only their ratio approaches $1$ near zero.

Answer 36 from the source-traced unit appendix.

Exercise 37

Give a squeeze argument for $\lim_{x\to0}x^4\sin(1/x^3)$ .

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Use $-x^4\le x^4\sin(1/x^3)\le x^4$ ; the limit is zero.

Answer 37 from the source-traced unit appendix.

Exercise 38

Create a trigonometric limit near zero whose value is $7/4$ .

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Example: $\lim_{x\to0}\sin(7x)/(4x)=7/4$ .

Answer 38 from the source-traced unit appendix.

Answers begin in the referenced section.

After the explanation

Use the section idea

Reading lens

Can the expression be rewritten around a known small-angle limit, with every scaling factor accounted for?

Mental model

The fundamental sine limit is a reusable local shape: other trigonometric limits work when you expose that shape through identities and scaling.

Decision

Look for a bounded oscillation times a shrinking factor, or rewrite the expression into sine-over-angle factors whose arguments match their denominators.

Common trap

The sine function is not equal to its angle; their ratio merely approaches one near zero, and that statement requires radian measure.

Check yourself

Mastery means you can mark every scaling factor before simplifying and can explain where the Squeeze Theorem enters the argument.

Source & rights

Original instruction with traceable references.

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