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

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\rightarrow\infty )

(\lim_{x\rightarrow\infty}\ln y=\lim_{x\rightarrow\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\rightarrow\infty}\frac{4\ln x}{x}=\lim_{x\rightarrow\infty}\frac{\frac{4}{x}}{1})

Step3: Evaluate the limit

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

Answer:

(1)