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Continuity — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsContinuity2 Marks
MCQ
Let $f: R \rightarrow R$ be defined by
$f(x)=\left\{\begin{array}{cc}\frac{\cos 3 x-\cos x}{x^2}, & x \neq 0 \\\lambda, & x=0\end{array} .\right.$
If f is continuous at $x=0$, then $\lambda$ is equal to
  • A
    -2
  • ✓
    -4
  • C
    -6
  • D
    -8
✓

Answer

Correct option: B.
-4
(B)
Since $f (x)$ is continuous $x=0$.
$\therefore \quad f (0)=\lim _{x \rightarrow 0} f (x)$
$\Rightarrow \lambda=\lim _{x \rightarrow 0} \frac{\cos 3 x-\cos x}{x^2}$
Applying L'Hospital rule on R.H.S., we get
$\lambda=\lim _{x \rightarrow 0} \frac{-3 \sin 3 x+\sin x}{2 x}$
Applying L'Hospital rule on R.H.S., we get
$\lambda=\lim _{x \rightarrow 0} \frac{-9 \cos 3 x+\cos x}{2} \Rightarrow \lambda=\frac{-9+1}{2}=-4$

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