Question:
If function $f(x) = \begin{cases} x\, \sin\left(\frac{1}{x} \right) ; & \text{x $\ne$ 0} \\[2ex] a \,;& \text{x = 0} \end{cases}$
is continuous at $x = 0$, then value of $a$ is
If function $f(x) = \begin{cases} x\, \sin\left(\frac{1}{x} \right) ; & \text{x $\ne$ 0} \\[2ex] a \,;& \text{x = 0} \end{cases}$
is continuous at $x = 0$, then value of $a$ is
Updated On: Jun 20, 2022
- $1$
- $-1$
- $0$
- None of the options
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The Correct Option is C
Solution and Explanation
We have, $f(0)=a$
$\therefore \displaystyle\lim _{x \rightarrow 0^{+}} f(x)=\displaystyle\lim _{h \rightarrow 0} f(0+h)=\displaystyle\lim _{h \rightarrow 0} h \sin \frac{1}{h}=0$
and $\displaystyle\lim _{x \rightarrow 0^{-}} f(x)=\displaystyle\lim _{h \rightarrow 0} f(0-h)$
$=\lim _{h \rightarrow 0}(-h) \sin \frac{1}{-h}=0$
Since, $f(x)$ is continuous at $x=0$, we must have
$f(0)=\displaystyle\lim _{x \rightarrow 0^{+}} f(x)=\displaystyle\lim _{x \rightarrow 0^{-}} f(x) \Rightarrow a=0$
$\therefore \displaystyle\lim _{x \rightarrow 0^{+}} f(x)=\displaystyle\lim _{h \rightarrow 0} f(0+h)=\displaystyle\lim _{h \rightarrow 0} h \sin \frac{1}{h}=0$
and $\displaystyle\lim _{x \rightarrow 0^{-}} f(x)=\displaystyle\lim _{h \rightarrow 0} f(0-h)$
$=\lim _{h \rightarrow 0}(-h) \sin \frac{1}{-h}=0$
Since, $f(x)$ is continuous at $x=0$, we must have
$f(0)=\displaystyle\lim _{x \rightarrow 0^{+}} f(x)=\displaystyle\lim _{x \rightarrow 0^{-}} f(x) \Rightarrow a=0$
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Concepts Used:
Limits
A function's limit is a number that a function reaches when its independent variable comes to a certain value. The value (say a) to which the function f(x) approaches casually as the independent variable x approaches casually a given value "A" denoted as f(x) = A.
If limx→a- f(x) is the expected value of f when x = a, given the values of ‘f’ near x to the left of ‘a’. This value is also called the left-hand limit of ‘f’ at a.
If limx→a+ f(x) is the expected value of f when x = a, given the values of ‘f’ near x to the right of ‘a’. This value is also called the right-hand limit of f(x) at a.
If the right-hand and left-hand limits concur, then it is referred to as a common value as the limit of f(x) at x = a and denote it by lim x→a f(x).