Does the following limits exist? If they exist find the value
$lim x \to 0 e^{1 / x}$

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does not exist

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Solution

The correct option is D does not exist
Let $f (x) = e^{1 / x} .$
Then $f (0 + 0) = lim h \to 0 e^{1 / (0 + h)} = lim h \to 0 = e^{(1 / h)} = e^{\infty} = \infty,$ and $f (0 - 0) = lim h \to 0 e^{1 / (0 - h)}$
$= lim h \to 0 e^{(- 1 / h)} = e^{- \infty} = \frac{1}{e^{\infty}} = \frac{1}{\infty} = 0.$
Since the right hand and the left hand limits are unequal, the $lim x \to 0 e^{1 / x}$ does not exist.
Ans: D

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