Question

1. find lim x→∞ x^(1/x).

Explanation:

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

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

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

Use L'Hopital's Rule. If we consider the limit \( \lim_{x\to\infty}\frac{\ln x}{x} \), which is in the \( \frac{\infty}{\infty} \) form.
By L'Hopital's Rule, \( \lim_{x\to\infty}\frac{\ln x}{x}=\lim_{x\to\infty}\frac{\frac{1}{x}}{1} \)

Step3: Evaluate the limit of \( \frac{1}{x} \) as \( x\to\infty \)

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

Step4: Find the limit of \( y \) as \( x\to\infty \)

\( \lim_{x\to\infty}y=\lim_{x\to\infty}e^{\ln y}=e^{\lim_{x\to\infty}\ln y} \)

Answer:

\( 1 \)