Trigonometric Limits

The basic trigonometric limit is

$\lim\limits_{x \to 0} \frac{{\sin x}}{x} = 1.$

Using this limit, one can get the series of other trigonometric limits:

${\lim\limits_{x \to 0} \frac{{\tan x}}{x} = 1,\;\;\;}\kern-0.3pt {\lim\limits_{x \to 0} \frac{{\arcsin x}}{x} = 1,\;\;\;}\kern-0.3pt {\lim\limits_{x \to 0} \frac{{\arctan x}}{x} = 1.}$

Further we assume that angles are measured in radians.

Solved Problems

Click or tap a problem to see the solution.

Example 1

Find the limit $$\lim\limits_{x \to 0} {\large{\frac{{4x}}{{\sin 3x}}}\normalsize}$$.

Example 2

Calculate the limit $$\lim\limits_{x \to 0} {\large\frac{{\cos {3x} – \cos x}}{{{x^2}}}\normalsize}$$.

Example 3

Find the limit $$\lim\limits_{x \to 0} {\large\frac{{\sin5x – \sin 3x}}{{\sin x}}\normalsize}$$.

Example 4

Calculate the limit $$\lim\limits_{x \to 0} {\large{\frac{{\cos \left( {x + a} \right) – \cos \left( {x – a} \right)}}{x}}\normalsize}.$$

Example 5

Calculate the limit $$\lim\limits_{x \to 0} {\large\frac{{\sin ax}}{{\sin bx}}\normalsize}$$.

Example 6

Find the limit $$\lim\limits_{x \to b} {\large\frac{{\sin x – \sin b}}{{x – b}}\normalsize}$$.

Example 7

Find the limit $$\lim\limits_{x \to 0} {\large\frac{{\tan x – \sin x}}{{{x^3}}}\normalsize}$$.

Example 8

Find the limit $$\lim\limits_{x \to {\large\frac{1}{2}\normalsize}} {\large\frac{{1 – 4{x^2}}}{{\arcsin \left( {1 – 2x} \right)}}\normalsize}$$.

Example 9

Find the limit $$\lim\limits_{x \to 0 + 0} {\large\frac{{\sqrt {1 – \cos x} }}{x}\normalsize}$$.

Example 1.

Find the limit $$\lim\limits_{x \to 0} {\large{\frac{{4x}}{{\sin 3x}}}\normalsize}$$.

Solution.

$L = {\lim\limits_{x \to 0} \frac{{4x}}{{\sin 3x}} } = {\lim\limits_{x \to 0} \frac{{3 \cdot 4x}}{{3\sin 3x}} } = {\frac{4}{3}\lim\limits_{x \to 0} \frac{{3x}}{{\sin 3x}} } = {\frac{4}{3}\lim\limits_{x \to 0} \frac{1}{{\large\frac{{\sin 3x}}{{3x}}\normalsize}} } = {\frac{4}{3}\frac{{\lim\limits_{x \to 0} 1}}{{\lim\limits_{x \to 0} \large\frac{{\sin 3x}}{{3x}}\normalsize}}.}$

Since $$3x \to 0$$ as $$x \to 0,$$ we can write:

$L = \frac{4}{3}\frac{{\lim\limits_{x \to 0} 1}}{{\lim\limits_{x \to 0} \large\frac{{\sin 3x}}{{3x}}\normalsize}} = \frac{4}{{3\lim\limits_{3x \to 0} \large\frac{{\sin 3x}}{{3x}}\normalsize}} = \frac{4}{{3 \cdot 1}} = \frac{4}{3}.$

Example 2.

Calculate the limit $$\lim\limits_{x \to 0} {\large\frac{{\cos {3x} – \cos x}}{{{x^2}}}\normalsize}$$.

Solution.

We factor the numerator:

${\cos{3x} – \cos x } = { – 2\sin \frac{{3x – x}}{2}\sin \frac{{3x + x}}{2} } = { – 2\sin x\sin 2x. }$

This yields

${\lim\limits_{x \to 0} \frac{{\cos 3x – \cos x}}{{{x^2}}} } = {\lim\limits_{x \to 0} \frac{{\left( { – 2\sin x\sin 2x} \right)}}{{{x^2}}} } = {- 2\lim\limits_{x \to 0} \frac{{\sin x}}{x} \cdot \lim\limits_{x \to 0} \frac{{\sin 2x}}{x} } = {- 2 \cdot 1 \cdot \lim\limits_{2x \to 0} \frac{{2\sin 2x}}{{2x}} } = {- 2 \cdot 2\lim\limits_{2x \to 0} \frac{{\sin 2x}}{{2x}} = – 4.}$

Example 3.

Find the limit $$\lim\limits_{x \to 0} {\large\frac{{\sin5x – \sin 3x}}{{\sin x}}\normalsize}$$.

Solution.

We use the following trigonometric identity:

${\sin x – \sin y }={ 2\sin \frac{{x – y}}{2}\cos \frac{{x + y}}{2}.}$

Then we obtain

${\lim\limits_{x \to 0} \frac{{\sin5x – \sin 3x}}{{\sin x}} } = {\lim\limits_{x \to 0} \frac{{2\sin \large\frac{{5x – 3x}}{2}\normalsize\cos \large\frac{{5x + 3x}}{2}\normalsize}}{{\sin x}} } = {\lim\limits_{x \to 0} \frac{{2\sin x\cos 4x}}{{\sin x}} } = {\lim\limits_{x \to 0} \left( {2\cos 4x} \right).}$

As $$\cos{4x}$$ is a continuous function at $$x = 0,$$ then

${\lim\limits_{x \to 0} \left( {2\cos 4x} \right) } = {2\lim\limits_{x \to 0} \cos 4x } = {2 \cdot \cos \left( {4 \cdot 0} \right) = 2 \cdot 1 = 2.}$

Page 1
Problems 1-3
Page 2
Problems 4-9