Macdonald Sherman
06/04/2024 · Senior High School
Find the limit. \[ \lim x^{\frac{4}{1-x}} \]
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Step-by-step Solution
Calculate the limit \[ \lim_{x\rightarrow 1} x^{\frac{4}{1-x}} \).
Evaluate the limit by following steps:
- step0: Evaluate using L'Hopital's rule:
\(\lim _{x\rightarrow 1}\left(x^{\frac{4}{1-x}}\right)\)
- step1: Transform the expression:
\(\lim _{x\rightarrow 1}\left(e^{\frac{4}{1-x}\times \ln{\left(x\right)}}\right)\)
- step2: Transform the expression:
\(e^{\lim _{x\rightarrow 1}\left(\frac{4}{1-x}\times \ln{\left(x\right)}\right)}\)
- step3: Use the L'Hopital's rule:
\(e^{-4}\)
- step4: Express with a positive exponent:
\(\frac{1}{e^{4}}\)
The limit of \( x^{\frac{4}{1-x}} \) as \( x \) approaches 1 is \( \frac{1}{e^{4}} \).
Quick Answer
The limit is \( \frac{1}{e^{4}} \).
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