How to Find Limit of Trigonometric Functions

Here, you will learn how to find limit of trigonometric functions and limits using series expansion with example.

Let’s begin – 

Limit of Trigonometric Functions

$\displaystyle{\lim_{x \to 0}}$ $sinx\over x$ = 1 = $\displaystyle{\lim_{x \to 0}}$ $tanx\over x$ = $\displaystyle{\lim_{x \to 0}}$ $tan^{-1}x\over x$ = $\displaystyle{\lim_{x \to 0}}$ $sin^{-1}x\over x$ [where x is measured in radians]

(a)  If $\displaystyle{\lim_{x \to a}}$ f(x) = 0, then $\displaystyle{\lim_{x \to a}}$ $sinf(x)\over f(x)$ = 1
e.g. $\displaystyle{\lim_{x \to 1}}$ $sin(lnx)\over (lnx)$ = 1 

Example : Evaluate : $\displaystyle{\lim_{x \to 0}}$ $x^3 cotx\over {1-cosx}$

Solution : $\displaystyle{\lim_{x \to 0}}$ $x^3 cosx\over {sinx(1-cosx)}$ = $\displaystyle{\lim_{x \to 0}}$ $x^3 cosx(1 + cosx)\over {sinxsin^2x}$ = $\displaystyle{\lim_{x \to 0}}$ ${x^3\over sin^3x}.cosx(1 + cosx)$ = 2

Example : Evaluate : $\displaystyle{\lim_{x \to 0}}$ $(2+x)sin(2+x)-2sin2\over x$

Solution : $\displaystyle{\lim_{x \to 0}}$ $2(sin(2+x)-sin2)+xsin(2+x)\over x$

= $\displaystyle{\lim_{x \to 0}}$($2.2.cos(2+{x\over 2})sin{x\over 2}\over x$ + sin(2+x))

= $\displaystyle{\lim_{x \to 0}}$$2cos(2+{x\over 2})sin{x\over 2}\over {x\over 2}$ + $\displaystyle{\lim_{x \to 0}}$sin(2+x)

= 2cos2 + sin2

Example : Evaluate : $\displaystyle{\lim_{x \to 0}}$ $xln(1+2tanx)\over 1-cosx$

Solution : $\displaystyle{\lim_{x \to 0}}$ $xln(1+2tanx)\over 1-cosx$

= $\displaystyle{\lim_{x \to 0}}$ $xln(1+2tanx)\over {1-cosx\over x^2}.x^2$.$2tanx\over 2tanx$

= 4

Limit using series expansion

Expansion of function like binomial expansion, exponential & logarithmic expansion, expansion of sinx, cosx, tanx should be remembered by heart which are given below :

(a)  $e^x$ = 1 + $x\over 1!$ + $x^2\over {2!}$ + ……..

(b)  ln(1 + x) = x – $x^2\over 2$ + $x^3\over 3$ – $x^4\over 4$ + ………for -1 < x $\leq$ 1

(c)  sinx = x – $x^3\over 3!$ + $x^5\over 5!$ – $x^7\over 7!$ + ……….

(d)  cosx = 1 – $x^2\over 2!$ + $x^4\over 4!$ + $x^6\over 6!$ + ……….

(e)  tanx = x + $x^3\over 3$ + $2x^5\over 15$ + …….

Hope you learnt how to find the limit of trigonometric functions. To learn more practice more questions and get ahead in competition. Good Luck!