The value that should be assigned to f(0) so | vedclass

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(A) For the function $f(x)$ to be continuous at $x=0$, we must have $f(0) = \lim_{x \rightarrow 0} f(x)$.Given $f(x) = (x+1)^{\cot x}$.Taking the limi

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$e$

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(A) For the function $f(x)$ to be continuous at $x=0$, we must have $f(0) = \lim_{x ightarrow 0} f(x)$.Given $f(x) = (x+1)^{\cot x}$.Taking the limit as $x ightarrow 0$:$\lim_{x ightarrow 0} (x+1)^{\cot x} = e^{\lim_{x ightarrow 0} \cot x \cdot \ln(1+x)}$.Using the standard limit $\lim_{x ightarrow 0} \frac{\ln(1+x)}{x} = 1$, we can rewrite the exponent as:$\lim_{x ightarrow 0} \frac{\ln(1+x)}{	an x} = \lim_{x ightarrow 0} \left( \frac{\ln(1+x)}{x} \cdot \frac{x}{	an x} ight)$.Since $\lim_{x ightarrow 0} \frac{\ln(1+x)}{x} = 1$ and $\lim_{x ightarrow 0} \frac{x}{	an x} = 1$, the limit of the exponent is $1 \cdot 1 = 1$.Therefore, $f(0) = e^1 = e$.

The value that should be assigned to $f(0)$ so that the function $f(x)=(x+1)^{\cot x}$ is continuous at $x=0$, is

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