find the limit. use lhospitals rule where appropriate. if there is a more elementary method, consider using…

find the limit. use lhospitals rule where appropriate. if there is a more elementary method, consider using it. \n lim _{x \rightarrow 0}(1-4 x)^{1 / x}

find the limit. use lhospitals rule where appropriate. if there is a more elementary method, consider using it. \n lim _{x \rightarrow 0}(1-4 x)^{1 / x}

Answer

Explanation:

Step1: Let ( y=(1 - 4x)^{\frac{1}{x}} )

Take the natural logarithm of both sides: ( \ln y=\frac{\ln(1 - 4x)}{x} )

Step2: Find the limit of ( \ln y ) as ( x\rightarrow0 )

As ( x\rightarrow0 ), we have the indeterminate form ( \frac{0}{0} ). Apply L'Hospital's Rule. Differentiate the numerator and denominator: The derivative of ( \ln(1 - 4x) ) is ( \frac{-4}{1 - 4x} ), and the derivative of ( x ) is ( 1 ). So ( \lim_{x\rightarrow0}\ln y=\lim_{x\rightarrow0}\frac{-4}{1 - 4x}=- 4 )

Step3: Find the limit of ( y )

Since ( y = e^{\ln y} ), then ( \lim_{x\rightarrow0}y=\lim_{x\rightarrow0}e^{\ln y}=e^{\lim_{x\rightarrow0}\ln y} ) Substitute ( \lim_{x\rightarrow0}\ln y=-4 )

Answer:

( e^{-4} )