CBSE 2024 · Region 5 · Set 2 · Q25 · 2 marks
Verify whether the function $\displaystyle \mathrm{f}$ defined by
is continuous at $\displaystyle x=0$ or not.Check for differentiability of the function f defined by $\displaystyle \mathrm{f}(x)=|x-5|$, at the point $\displaystyle x=5$.
Verify whether the function $\displaystyle \mathrm{f}$ defined by
is continuous at $\displaystyle x=0$ or not.
Check for differentiability of the function f defined by $\displaystyle \mathrm{f}(x)=|x-5|$, at the point $\displaystyle x=5$.
Marking-scheme solution
(a)
$\displaystyle \lim _{x \rightarrow 0} \mathrm{f}(x)=\lim _{x \rightarrow 0} x \cdot \sin \frac{1}{x}=0 \times$ Finite value in $\displaystyle [-1,1]=0=\mathrm{f}(0)$
$\displaystyle \therefore \mathbf{f}(\mathbf{x})$ is a continuous function.
Or
(b) LHD $\displaystyle =\lim _{x \rightarrow 5^{-}} \frac{|x-5|-0}{x-5}=\lim _{x \rightarrow 5^{-}} \frac{-(x-5)}{x-5}=-1$
$$\begin{aligned}
& \text { RHD }=\lim _{x \rightarrow $\displaystyle 5$^{+}} \frac{|x-5|-0}{x-5}=\lim _{x \rightarrow $\displaystyle 5$^{+}} \frac{(x-5)}{x-5}=$\displaystyle 1$
& \text { LHD } \neq \text { RHD, } \therefore \mathrm{f}(x) \text { is not differentiable at } x=$\displaystyle 5$
\end{aligned}
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CBSE Class 12 Mathematics past-paper question from the 2024board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.