evaluate the limit using lhospitals rule if necessary\n lim _{x \rightarrow infty}left(1+\frac{10}{x}\right)^…

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\rightarrow\infty}\ln y=\lim_{x\rightarrow\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\rightarrow\infty}\frac{\frac{1}{1+\frac{10}{x}}\times(-\frac{10}{x^{2}})}{-\frac{1}{x^{2}}}=\lim_{x\rightarrow\infty}\frac{10}{1+\frac{10}{x}})

Step4: Evaluate the limit

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

Step5: Find the original limit

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

Answer:

(e^{10})