$\frac{e^{1 / x} - e^{- 1 / x}}{e^{1 / x} + e^{- 1 / x}}$ check continuity at $x = 0^{-}$

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Solution

$f (x) = \frac{e^{1 / x} - e^{- 1 / x}}{e^{1 / x} + e^{- 1 / x}}$

$f (x) = 1 - \frac{2 e^{- 1 / x}}{e^{1 / x} + e^{- 1 / x}}$

$f (0) = 1 - 2 (0) = 1$

$f (0^{-}) = 1 - \frac{2 e^{- 1 / x}}{e^{1 / x} + e^{- 1 / x}}$

$= lim x \to 0 (1 - \frac{2 e^{1 / x}}{e^{1 / x} + e^{- 1 / x}})$

$= 1 - 2$

$= - 1$

$∴ D i s c o n t i n u o u s$

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