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!