In order that the function f(x) = (x + 1)^{co | vedclass

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(C) For the function $f(x)$ to be continuous at $x = 0$,we must have $f(0) = \lim_{x 	o 0} f(x)$.We evaluate the limit: $\lim_{x 	o 0} (x + 1)^{\cot

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$f(0) = e$

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(C) For the function $f(x)$ to be continuous at $x = 0$,we must have $f(0) = \lim_{x 	o 0} f(x)$.We evaluate the limit: $\lim_{x 	o 0} (x + 1)^{\cot x}$.This is an indeterminate form of type $1^\infty$. We can rewrite the expression as:$\lim_{x 	o 0} (1 + x)^{\cot x} = \lim_{x 	o 0} \left[ (1 + x)^{1/x} ight]^{x \cot x}$.We know that $\lim_{x 	o 0} (1 + x)^{1/x} = e$.Also,$\lim_{x 	o 0} x \cot x = \lim_{x 	o 0} \frac{x}{	an x} = 1$.Therefore,$\lim_{x 	o 0} f(x) = e^1 = e$.Thus,$f(0)$ must be defined as $e$ for the function to be continuous at $x = 0$.

In order that the function $f(x) = (x + 1)^{\cot x}$ is continuous at $x = 0$,$f(0)$ must be defined as

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