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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

limx0 sinxx = 1 = limx0 tanxx = limx0 tan1xx = limx0 sin1xx [where x is measured in radians]

(a)  If limxa f(x) = 0, then limxa sinf(x)f(x) = 1
e.g. limx1 sin(lnx)(lnx) = 1 

Example : Evaluate : limx0 x3cotx1cosx

Solution : limx0 x3cosxsinx(1cosx) = limx0 x3cosx(1+cosx)sinxsin2x = limx0 x3sin3x.cosx(1+cosx) = 2

Example : Evaluate : limx0 (2+x)sin(2+x)2sin2x

Solution : limx0 2(sin(2+x)sin2)+xsin(2+x)x

= limx0(2.2.cos(2+x2)sinx2x + sin(2+x))

= limx02cos(2+x2)sinx2x2 + limx0sin(2+x)

= 2cos2 + sin2

Example : Evaluate : limx0 xln(1+2tanx)1cosx

Solution : limx0 xln(1+2tanx)1cosx

= limx0 xln(1+2tanx)1cosxx2.x2.2tanx2tanx

= 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)  ex = 1 + x1! + x22! + ……..

(b)  ln(1 + x) = x – x22 + x33x44 + ………for -1 < x 1

(c)  sinx = x – x33! + x55!x77! + ……….

(d)  cosx = 1 – x22! + x44! + x66! + ……….

(e)  tanx = x + x33 + 2x515 + …….

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

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