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!
