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Continuity — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsContinuity3 Marks
Question
Show that the following functions have a continuous extension to the point where $f(x)$ is not defined. Also, find the extension:$f ( x )=\frac{1-\cos 2 x}{\sin x}$, for $x \neq 0$.
✓

Answer

$f(x)=\frac{1-\cos 2 x}{\sin x} \text {, for } x \neq 0$
Here, $f(0)$ is not defined.
Consider,
$\lim _{x \rightarrow 0} f(x) =\lim _{x \rightarrow 0} \frac{1-\cos 2 x}{\sin x}$
$ =\lim _{x \rightarrow 0} \frac{2 \sin ^2 x}{\sin x}$
$=2 \lim _{x \rightarrow 0}(\sin x) \quad \ldots\left[\begin{array}{l} \because x \rightarrow 0, x \neq 0 \\ \therefore \sin x \neq 0 \end{array}\right]$
$=2(\sin 0)=2 \times 0$
$ =0$
$\lim _{x \rightarrow 0} f(x)$ exists.
But $f(0)$ is not defined.
$\therefore f ( x )$ has a removable discontinuity at $x =0$.
$\therefore$ The extension of the original function is
$f(x)=\frac{1-\cos 2 x}{\sin x}$ for $x \neq 0$
$=0$ for $x=0$
$\therefore f ( x )$ is continuous at $x =0$.

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