Calculate Limits of Trigonometric Functions

Several examples related to the limits of trigonometric functions with detailed solutions and exercises with answers are presented.

Examples and Solutions

Example 1

Find the limit \[ \lim_{x \to 0} \frac{1 - \cos x}{x} \] Solution to Example 1:
Let us multiply the numerator and denominator by $ 1 + \cos x $ and write \[ \lim_{x \to 0} \frac{1 - \cos x}{x} = \lim_{x \to 0} \frac{1 - \cos x}{x} \cdot \frac{1 + \cos x}{1 + \cos x} \] The numerator becomes equal to $ 1 - \cos^2 x = \sin^2 x $, hence \[ \lim_{x \to 0} \frac{1 - \cos x}{x} = \lim_{x \to 0} \frac{\sin^2 x}{x} \cdot \frac{1}{1 + \cos x} \] The limit can be written \[ \lim_{x \to 0} \frac{1 - \cos x}{x} = \lim_{x \to 0} \left( \frac{\sin x}{x} \right) \cdot \lim_{x \to 0} \left( \frac{\sin x}{1 + \cos x} \right) = (1)\left(\frac{0}{2}\right) = 0 \] We have used the theorem: $\lim_{x \to 0} \dfrac{\sin x}{x} = 1$.

Example 2

Find the limit \[ \lim_{x \to 0} \dfrac{\sin 4 x}{4 x} \]
Solution to Example 2:
Let $ t = 4 x $. When $ x $ approaches 0, $t$ approaches 0, so that \[ \lim_{x \to 0} \dfrac {\sin 4 x}{4 x} = \lim_{t \to 0} \dfrac {\sin t}{t} \] We now use the theorem: $ \lim_{t \to 0} \dfrac {\sin t}{t} = 1 $ to find the limit \[ \lim_{x \to 0} \dfrac {\sin 4 x}{4 x} = \lim_{t \to 0} \dfrac {\sin t}{t} = 1 \]

Example 3

Find the limit \[ \lim_{t \to 0} \dfrac{\sin 6 x}{5 x} \] Solution to Example 3:
Let $ t = 6 x $ or $ x = t / 6 $. When $ x $ approaches 0, $ t $ approaches 0, so that \[ \lim_{t \to 0} \dfrac{\sin 6 x}{5 x} = \lim_{t \to 0} \dfrac{\sin t}{5 t/6} \] \[ = \lim_{t \to 0} (6 / 5) \dfrac {\sin t}{t} \] \[ = (6 / 5) \lim_{t \to 0} \dfrac {\sin t}{t} \] \[ = (6 / 5) \cdot 1 = 6 / 5 \]

Example 4

Find the limit \[ \lim_{x \to -3} \dfrac{\sin (x + 3)}{x^2 +7x + 12} \]
Solution to Example 4:
If we apply the theorem of the limit of the quotient of two functions, we will get the indeterminate form $ \dfrac{0}{0} $. We need to find another way. For $ x = -3 $, the denominator is equal to zero and therefore may be factorized, hence \[ \lim_{x \to -3} \dfrac{\sin (x + 3)}{x^2 +7x + 12} \] \[ = \lim_{x \to -3} \dfrac{\sin (x + 3)}{(x + 3)(x + 4)} \] Let $ t = x + 3 $ or $ x = t - 3 $. As $ x $ approaches $ -3 $, $ t $ approaches 0. \[ \lim_{x \to -3} \dfrac{\sin (x + 3)}{x^2+7x + 12} \] \[ = \lim_{t \to 0} \dfrac {\sin t} { t (t + 1) }\] We now apply the theorem of the limit of the product of two functions. \[ = \lim_{t \to 0} \dfrac{\sin t}{t} \cdot \lim_{t \to 0} \dfrac{1}{t+1} \] \[ = 1 \cdot 1 = 1 \]

Example 5

Find the limit \[ \lim_{x \to 0} \dfrac{\sin|x|}{x} \]
Solution to Example 5:
We shall find the limit as $x$ approaches 0 from the left and as $ x $ approaches $0$ from the right. For $ x \lt 0 $, $ | x | = -x $ \[ \lim_{x \to 0^-} \dfrac{\sin|x|}{x} \] \[ = \lim_{x \to 0^-} \dfrac{\sin (- x)}{x} \] \[ = - \lim_{x \to 0^-} \dfrac{\sin x}{x} \] \[ = -1 \] For $ x \gt 0 $, $ | x | = x $ \[ \lim_{x \to 0^+} \dfrac{\sin | x |}{x} \] \[ \lim_{x \to 0^+} \dfrac{\sin x }{x} \] \[ = 1 \] The limits from the left and from the right have different values, therefore the above limit does not exist. \[ \lim_{x \to 0} \dfrac{\sin | x |}{x} \quad \text{DOES NOT EXIST} \]

Example 6

Find the limit \[ \lim_{x \to 0} \dfrac{x}{\tan x} \]
Solution to Example 6:
We first use the trigonometric identity $ \tan x = \dfrac{\sin x}{\cos x} $ \[ \lim_{x \to 0} \dfrac{x}{\tan x} \] \[ = \lim_{x \to 0} \dfrac{x}{\dfrac{\sin x}{\cos x}} \] \[ = \lim_{x \to 0} \dfrac{ x \cos x}{\sin x} \] \[ = \lim_{x \to 0} \dfrac{\cos x}{\sin x / x} \] We now use the theorem of the limit of the quotient. \[ = \dfrac{\lim_{x \to 0} \cos x}{\lim_{x \to 0} (\sin x / x)} = 1 / 1 = 1 \]

Example 7

Find the limit \[ \lim_{x \to 0} x \csc x \] Solution to Example 7:
We first use the trigonometric identity $ \csc x = 1 / \sin x $ \[ \lim_{x \to 0} x \csc x \]
\[ = \lim_{x \to 0} x / \sin x \]
\[ = \lim_{x \to 0} \dfrac{1}{\sin x / x} \]
The limit of the quotient is used.
\[ = 1 / 1 = 1 \]

Exercises:

Find the limits
1. $ \lim_{x \to 0} \dfrac{\sin 3 x }{\sin 8 x} $
2. $ \lim_{x \to 0} \dfrac{\tan 3x}{x} $
3. $ \lim_{x \to 0} \sqrt x \, \csc ( 4 \sqrt x ) $
4. $ \lim_{x \to 0} \dfrac{\sin^3 3x}{x \, \sin(x^2)} $

Solutions to Above Exercises

Find the limits
1. 3/8
2. 3
3. 1/4
4. 27