If f(x) = {[x/sin3x,x ≠ 0][k, x = 0] is continuous at x = 0, then write the value of k. ​

Question

If  \(f(x) = \begin{cases} \frac{x}{sin3x}, & x ≠{0}\\ k, & x={ 0} \end{cases} \) is continuous at x = 0, then write the value of k.

Answer 1

Formula :- 

(i) \(\lim\limits_{x \to 0}\frac{sinx}{x}=1\)

(ii) 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)

\(\lim\limits_{x \to a}f(x)\) =  f(0)

Given :- 

\(f(x) = \begin{cases} \frac{x}{sin3x}, & x ≠{0}\\ k, & x={ 0} \end{cases} \) 

 \(\lim\limits_{x \to a}f(x)\) =  f(0)

\(\lim\limits_{x \to 0}\frac{x}{sin3x}\) = k

\(\frac{1}{3}\) = k