15.8. Limit of a Trigonometric Function

The limits of trigonometric functions have the following properties:

$1)\lim_{x\rightarrow c}\textup{sin}\, \:x=\textup{sin}\, \: c$

$2)\lim_{x\rightarrow c}\textup{cos}\, \:x=\textup{cos}\, \: c$

$3)\lim_{x\rightarrow c}\textup{tan}\, \:x=\textup{tan}\, \: c$

$4)\lim_{x\rightarrow c}\textup{csc}\, \:x=\textup{csc}\, \: c$

$5)\lim_{x\rightarrow c}\textup{sec}\, \:x=\textup{sec}\, \: c$

$6)\lim_{x\rightarrow c}\textup{cot}\, \:x=\textup{cot}\, \: c$

$7)\lim_{x\rightarrow 0}\frac{sin\, \:x}{x}=1$

$8)\lim_{x\rightarrow 0}\frac{1-cos\, \:x}{x}=0$

Except for #7 and #8, the limit of a trigonometric function can be found by plugging in c. #7 and #8 are examples of indeterminate forms. For these properties, we can use the more appropriate method or L'Hôpital's rule for limits.

Illustrative Example

Calculate the limit of 2 cos x + 2 as x approaches 3π / 4.
Using the 2nd property, we can plug 3π / 4 into cosx as follows:

$2\:cos\: x+2=2\:cos\: \frac{3\pi }{4}+2$

$\: \:\:\:=2\:\: (\frac{-\sqrt{2} }{2})+2$

$=-\sqrt{2}+2$

= 0.585786437626905
Therefore

$\lim_{x\rightarrow \frac{3\pi }{4}}\left ( 2\,\textup{cos}\, \, x +2\right )\approx 0.585786$

Calculator Solution
Type in: lim [ x = 3π / 4 ] ( 2 cos x + 2 )

More Examples

Calculate the limit of each trigonometric function if it exists.

$1)\lim_{x\rightarrow 0}\left ( x^2\, \textup{cos}\, \frac{1}{x} \right )$

Calculator solution
Type in: lim [ x = 0 ] ( x^2 cos( 1 / x ) )

$2) \lim_{\theta \rightarrow 0}(\frac{1-cos\theta }{\theta })$

Calculator solution
Type in: lim [ θ = 0 ] ( ( 1 - cosθ ) / θ )

$3) \lim_{x\rightharpoonup 1}\frac{\textup{cos}(x^2-1)-1}{x^2-1}$

Calculator solution
Type in: lim [ x = 1 ] ( ( cos( x^2 -1 ) -1 ) / ( x^2 - 1 ) )

Other Examples

Limit to Infinity

Some trigonometric functions such as tan, cot, sec and csc increase or decrease to positive or negative infinity respectively as x approaches c.

Illustrative Example
Calculate the limit of the given function below.

$\lim_{x\rightarrow \frac{3\pi }{2}^-}(\textup{tan}\: x+2)$

As x approaches 3π/2 from the left, the value of tan x increases to positive infinity such that the limit is ∞.

Calculator Solution
Type in: lim [ x = 3π/2 - ] ( tan x + 2 )