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  4. Limits that lead to the indeterminate forms 1∞, 00, ∞0 can sometimes be solved taking logarithm first and then using L'Hõpital 's rule Let displaystyle limx arrow a(f(x))g(x) is in the form of ∞0, it can be written as e displaystyle lim x arrow a g(x) ℓ n f(x)=eL where L= displaystyle lim x arrow a (ℓ nf(x)/1 / g(x)) is (∞/∞) form and can be solved using L'Hõpital 's rule. undersetx arrow 1+ textLim x1 /(1-x)

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Q. Limits that lead to the indeterminate forms $1^{\infty}, 0^0, \infty^0$ can sometimes be solved taking logarithm first and then using L'Hõpital 's rule
Let $\displaystyle\lim_{x \rightarrow a}(f(x))^{g(x)}$ is in the form of $\infty^0$, it can be written as $e^{\displaystyle\lim _{x \rightarrow a} g(x) \ell n f(x)}=e^L$
where $L=\displaystyle\lim _{x \rightarrow a} \frac{{\ell nf}(x)}{1 / g(x)}$ is $\frac{\infty}{\infty}$ form and can be solved using L'Hõpital 's rule.
$\underset{x \rightarrow 1^{+}}{\text{Lim}} x^{1 /(1-x)}$

Continuity and Differentiability

Solution:

Correct answer is (b) $e^{-1}$