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
limx→0 sinxx = 1 = limx→0 tanxx = limx→0 tan−1xx = limx→0 sin−1xx [where x is measured in radians]
(a) If limx→a f(x) = 0, then limx→a sinf(x)f(x) = 1
e.g. limx→1 sin(lnx)(lnx) = 1
Example : Evaluate : limx→0 x3cotx1−cosx
Solution :
limx→0 x3cosxsinx(1−cosx) =
limx→0 x3cosx(1+cosx)sinxsin2x =
limx→0 x3sin3x.cosx(1+cosx) = 2
Example : Evaluate : limx→0 (2+x)sin(2+x)−2sin2x
Solution : limx→0 2(sin(2+x)−sin2)+xsin(2+x)x
= limx→0(2.2.cos(2+x2)sinx2x + sin(2+x))
= limx→02cos(2+x2)sinx2x2 + limx→0sin(2+x)
= 2cos2 + sin2
Example : Evaluate : limx→0 xln(1+2tanx)1−cosx
Solution : limx→0 xln(1+2tanx)1−cosx
= limx→0 xln(1+2tanx)1−cosxx2.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 + x33 – x44 + ………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!