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
 Limit of (1-cos x)/x as x Approaches 0
Solution to Example 1:
Let us multiply the numerator and denominator by  1 + cos x and write
Limit Step 1
The numerator becomes is equal to Limit Step 2, hence
Limit Step 3
The limit can be written
Limit Step 4
We have used the theorem: Limit Theorem.



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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.
\( \lim_{x \to 0} \dfrac{\sin | x |}{x} \) 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


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