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!

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