If $f:R ightarrow R$ is defined by $f(x) = \left\{ \begin{array}{r} \frac{\cos{3x - \cos x}}{x^{2}},\ \ \ \ for\ x
eq 0 \ \lambda,\ \ \ \ \ \ \ \ \ for\ x = 0 \end{array} ight.\$ and if $f$ is continuous at $x = 0$, then $\lambda$ is equal to

Question

If $f:R ightarrow R$ is defined by $f(x) = \left\{ \begin{array}{r} \frac{\cos{3x - \cos x}}{x^{2}},\ \ \ \ for\ x 
eq 0 \ \lambda,\ \ \ \ \ \ \ \ \ for\ x = 0 \end{array} ight.\$ and if $f$ is continuous at $x = 0$, then $\lambda$ is equal to

Dr. Abubacker · Accepted Answer

(b)
LHL$= \underset{h \rightarrow 0}{lim}\frac{\cos{3(0 - h) - \cos{(0 - h)}}}{(0 - h)^{2}}$
$= \underset{h \rightarrow 0}{lim}\frac{\cos{3h - \cos h}}{h^{2}}$
$= \underset{h \rightarrow 0}{lim}\frac{- 3\sin{3h + \sin h}}{2h}$
$= \underset{h \rightarrow 0}{lim}\frac{{- 9\ cos}{3h + \cos h}}{2} = \frac{- 9 + 1}{2} = - 4$
$\because\ \ \lim_{x \rightarrow 0^{-}}{f(x) = f(0)\ \ \  \Rightarrow \ \ \ \ \lambda = - 4}$

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

Explanation

Reviewed by Dr. Abubacker

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