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Question

# Indeterminate Forms Show that the indeterminate forms $0^0$, $\infty^0$, and $1^{\infty}$ do not always have a value of 1 by evaluating each limit.$\lim\limits_{x \rightarrow 0}(x+1)^{(\ln 2) / x}$

Solution

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To solve this problem, we will first find the limit of $\left( x+1\right)^{ \frac{\ln2}{x}}$ as $x$ approaches zero using L'Hospital's Rule and direct substitution.

We will then use the fact that

$\lim_{x\to c} f(g(x)) = f\left(\lim_{x\to c}g(x)\right) \tag1$

if $f$ is continuous to find the final answer.

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