Question

evaluate the limit using lhospitals rule if necessary
lim _{x
ightarrow infty}left(1+\frac{10}{x}
ight)^{\frac{x}{10}}

Explanation:

Step1: Let \( y=(1 + \frac{10}{x})^{\frac{x}{1}} \), take the natural logarithm

\(\ln y=\frac{x}{1}\ln(1 + \frac{10}{x})\)

Step2: Rewrite the limit

\(\lim_{x
ightarrow\infty}\ln y=\lim_{x
ightarrow\infty}\frac{\ln(1+\frac{10}{x})}{\frac{1}{x}}\)
This is in the \(\frac{0}{0}\) form.

Step3: Apply L'Hospital's Rule

Differentiate the numerator and denominator.
The derivative of the numerator: \(\frac{d}{dx}\ln(1 + \frac{10}{x})=\frac{1}{1+\frac{10}{x}}\times(-\frac{10}{x^{2}})\)
The derivative of the denominator: \(\frac{d}{dx}\frac{1}{x}=-\frac{1}{x^{2}}\)
So \(\lim_{x
ightarrow\infty}\frac{\frac{1}{1+\frac{10}{x}}\times(-\frac{10}{x^{2}})}{-\frac{1}{x^{2}}}=\lim_{x
ightarrow\infty}\frac{10}{1+\frac{10}{x}}\)

Step4: Evaluate the limit

As \(x
ightarrow\infty\), \(\lim_{x
ightarrow\infty}\frac{10}{1+\frac{10}{x}} = 10\)
Since \(\lim_{x
ightarrow\infty}\ln y = 10\), and \(y = e^{\ln y}\)

Step5: Find the original limit

\(\lim_{x
ightarrow\infty}(1+\frac{10}{x})^{\frac{x}{1}}=e^{10}\)

Answer:

\(e^{10}\)