Question

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

lim_{x→∞}x^{4/x}

Explanation:

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

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

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

ightarrow\infty \)
\(\lim_{x
ightarrow\infty}\ln y=\lim_{x
ightarrow\infty}\frac{4\ln x}{x}\)
This is in the \(\frac{\infty}{\infty}\) form. Apply L'Hospital's Rule.
Differentiate the numerator and denominator: \(\lim_{x
ightarrow\infty}\frac{4\ln x}{x}=\lim_{x
ightarrow\infty}\frac{\frac{4}{x}}{1}\)

Step3: Evaluate the limit

\(\lim_{x
ightarrow\infty}\frac{\frac{4}{x}}{1}=0\)
Since \(\lim_{x
ightarrow\infty}\ln y = 0\), and \(y = e^{\ln y}\)
\(\lim_{x
ightarrow\infty}y=\lim_{x
ightarrow\infty}e^{\ln y}=e^{0}\)

Answer:

\(1\)