First slide

Continuity

Question

$If f (x) = (x + 1)^{\cot x} is continuous at x = 0, then f (0) is$

Moderate

A

1
B

e
C

$\frac{1}{e}$
D

none of these.

Solution

$\begin{array}{l} \lim_{x \to 0} f (x) = \lim_{x \to 0} {(1 + x)}^{^{\frac{1}{x} . x \cot x}} = e^{\lim_{x \to 0} x \cot x} \\ = e^{\lim_{x \to 0} \frac{x}{\tan x}} = e^{1} = e \\ S i n c e f (x) i s c o n t i n u o u s a t x = 0, ∴ f (0) = e . \end{array}$

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