(C) function $f(x)$ is continuous at $x = a$ if $\lim_{x \to a} f(x) = f(a)$.For option $C$,we check the limit at $x = 0$:$R.H.L. = \lim_{x \to 0^+} \

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$f(x) = \begin{cases} \frac{e^{1/x} - 1}{e^{1/x} + 1}, & x \neq 0 \\ -1, & x = 0 \end{cases}$

(C) function $f(x)$ is continuous at $x = a$ if $\lim_{x \to a} f(x) = f(a)$.For option $C$,we check the limit at $x = 0$:$R.H.L. = \lim_{x \to 0^+} \frac{e^{1/x} - 1}{e^{1/x} + 1} = \lim_{x \to 0^+} \frac{1 - e^{-1/x}}{1 + e^{-1/x}} = \frac{1 - 0}{1 + 0} = 1$.$L.H.L. = \lim_{x \to 0^-} \frac{e^{1/x} - 1}{e^{1/x} + 1} = \frac{0 - 1}{0 + 1} = -1$.Since $L.H.L. \neq R.H.L.$,the limit does not exist at $x = 0$.Therefore,the function is not continuous at $x = 0$.

Class 12 Mathematics Continuity and Differentiation Continuity

Easy

Which of the following functions is not continuous at $x = 0$?

MHT CET 2019 All MHT CET

A
$f(x) = \begin{cases} (1 + 2x)^{1/x}, & x \neq 0 \\ e^2, & x = 0 \end{cases}$
B
$f(x) = \begin{cases} \sin x - \cos x, & x \neq 0 \\ -1, & x = 0 \end{cases}$
C
$f(x) = \begin{cases} \frac{e^{1/x} - 1}{e^{1/x} + 1}, & x \neq 0 \\ -1, & x = 0 \end{cases}$
D
$f(x) = \begin{cases} \frac{e^{5x} - e^{2x}}{\sin 3x}, & x \neq 0 \\ 1, & x = 0 \end{cases}$

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Which of the following functions is not continuous at $x = 0$?

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