First slide

Continuity

Question

The value of $k$ for which the function $f (x) = \{\begin{cases} \begin{matrix} \sin \frac{1}{x} & x \neq 0 \end{matrix} \\ \begin{matrix} k & x = 0 \end{matrix} \end{cases}$ is continuous is

Easy

A

8
B

1
C

$- 1$
D

None of these

Solution

Since, $f (x) = \{\begin{cases} \begin{matrix} \sin \frac{1}{x} & x \neq 0 \end{matrix} \\ \begin{matrix} k & x = 0 \end{matrix} \end{cases}$
If $f (x)$ is continuous at $x = 0$
$\lim_{x \to 0} f (x) = f (0) = k$
$\Rightarrow \lim_{x \to 0} \sin \frac{1}{x} = k$
$\Rightarrow k = \sin \infty$ can take any finite value between $- 1$ and 1
$∴ f (x)$ is discontinuous at $x = 0$ .

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