If f(x) = {[asinπ/2(x+1),x≤1][(tan x -sin x)/x^3,x>0] is continuous at x = 0, then a equals : ​A. 1/2 ,B. 1/3

Question

If \(f(x) = \begin{cases} a\,sin\frac{\pi}{2}(x+1) &,x \leq{1}\\ \frac{tan\,x-sin\,x}{x^3} &, x>{ 0} \end{cases} \) is continuous at x = 0, then a equals :

A. \(\frac{1}{2}\)

B. \(\frac{1}{3}\)

C. \(\frac{1}{4}\)

D. \(\frac{1}{6}\)

Answer 1

Option : (A)

Formula :-

(i) A function f(x) is said to be continuous at a point x = a of its domain, if
\(\lim\limits_{x \to a}f(x)\) = f(a)
\(\lim\limits_{x \to a^+}f(a+h)\) = \(\lim\limits_{x \to a^-}f(a-h)\) = f(a)

Given :-

\(f(x) = \begin{cases} a\,sin\frac{\pi}{2}(x+1) &,x \leq{1}\\ \frac{tan\,x-sin\,x}{x^3} &, x>{ 0} \end{cases} \) 

Function f(x) is continuous at x = 0