lhopital: problem 1\n(1 point)\nevaluate the limit using lhospitals rule if necessary\n\\( \\lim _ { x…

Explanation:

Step1: Let ( y=\left(1 + \frac{10}{x}\right)^{\frac{x}{7}} )

Take the natural logarithm of both sides: ( \ln y=\frac{x}{7}\ln\left(1+\frac{10}{x}\right)=\frac{\ln\left(1 + \frac{10}{x}\right)}{\frac{7}{x}} )

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

As ( x\to\infty ), we have the indeterminate form ( \frac{0}{0} ). Apply L'Hospital's rule. Differentiate the numerator: ( \frac{d}{dx}\ln\left(1+\frac{10}{x}\right)=\frac{-\frac{10}{x^{2}}}{1+\frac{10}{x}}=\frac{- 10}{x(x + 10)} ) Differentiate the denominator: ( \frac{d}{dx}\frac{7}{x}=-\frac{7}{x^{2}} ) Then ( \lim_{x\to\infty}\frac{\ln\left(1+\frac{10}{x}\right)}{\frac{7}{x}}=\lim_{x\to\infty}\frac{\frac{-10}{x(x + 10)}}{-\frac{7}{x^{2}}}=\lim_{x\to\infty}\frac{10x^{2}}{7x(x + 10)}=\lim_{x\to\infty}\frac{10x}{7(x + 10)} ) Again, as ( x\to\infty ), we can divide numerator and denominator by ( x ): ( \lim_{x\to\infty}\frac{10x}{7(x + 10)}=\frac{10}{7} )

Step3: Find the limit of ( y )

Since ( \lim_{x\to\infty}\ln y=\frac{10}{7} ), and ( y = e^{\ln y} ), then ( \lim_{x\to\infty}y=e^{\frac{10}{7}} )

Answer:

( e^{\frac{10}{7}} )