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

find the limit. use lhospitals rule where appropriate. if there is a more elementary method, consider using it.\n lim _ { x \rightarrow infty } left( 1 + \frac { a } { x } \right) ^ { b x }

find the limit. use lhospitals rule where appropriate. if there is a more elementary method, consider using it.\n lim _ { x \rightarrow infty } left( 1 + \frac { a } { x } \right) ^ { b x }

Answer

Explanation:

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

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

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

We have ( \lim_{x\rightarrow\infty}\ln y=\lim_{x\rightarrow\infty}bx\ln\left(1 + \frac{a}{x}\right) ). Let ( t=\frac{1}{x} ), then as ( x\rightarrow\infty ), ( t\rightarrow0 ). The limit becomes ( \lim_{t\rightarrow0}\frac{b\ln(1 + at)}{t} ). This is in the ( \frac{0}{0} ) form. By L'Hospital's Rule, ( \lim_{t\rightarrow0}\frac{b\ln(1 + at)}{t}=\lim_{t\rightarrow0}\frac{b\times\frac{a}{1+at}}{1}=ab )

Step3: Find the limit of ( y )

Since ( \lim_{x\rightarrow\infty}\ln y = ab ), and ( y = e^{\ln y} ), then ( \lim_{x\rightarrow\infty}y=e^{ab} )

Answer:

( e^{ab} )