Question

If $f\left(x\right)=\{\begin{array}{cc}\frac{\sin 3x}{x^2+x},& x\ne 0\\ \lambda +2,& x=0\end{array}$ is continuous at $x=0,$ then the value of $\lambda$ is

[2 marks]

• A. $0$
• B. $-1$
• C. $1$
• D. $3$

Answer 1

The correct option is C $1$

For continuity at $x=0$, we must have
$f\left(0\right)=\underset{x\to 0}{lim}f\left(x\right)=\underset{x\to 0}{lim}\frac{\sin 3x}{x^2+x}$
$\Rightarrow f\left(0\right)=\underset{x\to 0}{lim}\frac{3x\cdot \frac{\sin 3x}{3x}}{x^2+x}$
$\Rightarrow f\left(0\right)=\underset{x\to 0}{lim}\frac{3x}{x^2+x}$
$\Rightarrow f\left(0\right)=\underset{x\to 0}{lim}\frac{3}{x+1}$
$\Rightarrow f\left(0\right)=3$
$\Rightarrow \lambda +2=3$
$\therefore \lambda =1$