Schneider Gross
01/26/2023 · Elementary School
Usar regla de LhopitaL \( \lim _{x \rightarrow 1}\left(\frac{x \ln (x)-(x-1)}{(x-1) \ln (x)}\right)^{2} \)
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Step-by-step Solution
Calculate the limit \( \lim _{x \rightarrow 1}\left(\frac{x \ln (x)-(x-1)}{(x-1) \ln (x)}\right)^{2} \).
Evaluate the limit by following steps:
- step0: Evaluate using L'Hopital's rule:
\(\lim _{x\rightarrow 1}\left(\left(\frac{x\ln{\left(x\right)}-\left(x-1\right)}{\left(x-1\right)\times \ln{\left(x\right)}}\right)^{2}\right)\)
- step1: Remove the parentheses:
\(\lim _{x\rightarrow 1}\left(\left(\frac{x\ln{\left(x\right)}-x+1}{\left(x-1\right)\times \ln{\left(x\right)}}\right)^{2}\right)\)
- step2: Rewrite the expression:
\(\left(\lim _{x\rightarrow 1}\left(\frac{x\ln{\left(x\right)}-x+1}{\left(x-1\right)\times \ln{\left(x\right)}}\right)\right)^{2}\)
- step3: Calculate:
\(\left(\frac{1}{2}\right)^{2}\)
- step4: Use the properties of exponents:
\(\frac{1^{2}}{2^{2}}\)
- step5: Evaluate the power:
\(\frac{1}{2^{2}}\)
- step6: Evaluate the power:
\(\frac{1}{4}\)
La expresión dada es \( \lim _{x \rightarrow 1}\left(\frac{x \ln (x)-(x-1)}{(x-1) \ln (x)}\right)^{2} \).
Al calcular el límite, obtenemos que el valor es \( \frac{1}{4} \) o 0.25.
Quick Answer
El valor del límite es \( \frac{1}{4} \) o 0.25.
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