Question

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

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

ightarrow0 \)
As \( x
ightarrow0 \), 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 \( \lim_{x
ightarrow0}\ln y=\lim_{x
ightarrow0}\frac{-4}{1 - 4x}=- 4 \)

Step3: Find the limit of \( y \)

Since \( y = e^{\ln y} \), then \( \lim_{x
ightarrow0}y=\lim_{x
ightarrow0}e^{\ln y}=e^{\lim_{x
ightarrow0}\ln y} \)
Substitute \( \lim_{x
ightarrow0}\ln y=-4 \)

Answer:

\( e^{-4} \)