Question

If $f : R \rightarrow R$ is defined by $f(x) = \begin{cases} \frac{\cos \ 3x - \cos \ x}{x^2} &, \text{for } x \neq 0 \\\ \lambda &, \text{for } x = \end{cases}$ and if $f$ is continuous at $x = 0,$ then $\lambda$ is equal to

• A. -2
• B. -4
• C. -6
• D. -8

Answer 1

Answer: (B)

-4

Answer 2

The correct option is(B): -4

Given that ,
$f(x) = \begin{cases} \frac{\cos \ 3x - \cos \ x}{x^2} &, \text{for } x \neq 0 \\\ \lambda &, \text{for } x = \end{cases}$
Now, LHL $=\displaystyle\lim _{x \rightarrow 0^{-}} f(x)$
$=\displaystyle\lim _{x \rightarrow 0^{-}} \frac{\cos 3 x-\cos x}{x^{2}}$
$=\displaystyle\lim _{h \rightarrow 0} \frac{\cos 3(0-h)-\cos 0-h}{0-h^{2}}$
$=\displaystyle\lim _{h \rightarrow 0} \frac{\cos 3 h-\cos h}{h^{2}}$
$=\displaystyle\lim _{h \rightarrow 0} \frac{-3 \sin 3 h+\sin h}{2 h}$ (using L' Hospitals' rule)
$=\frac{-9+1}{2}=-4$
Since, $f(x)$ is continuous at $x=0$
$\therefore \displaystyle\lim _{x \rightarrow 0^{-}} f(x)=f(0)$
$\Rightarrow-4=\lambda$
$\Rightarrow \lambda=-4$