Question-37

Consider two functions \(\displaystyle f( x)\) and \(\displaystyle g( x)\) such that \(\displaystyle \lim\limits _{x\rightarrow 5} \ f( x) =8\) and \(\displaystyle g( x) =2^{x}\). Let a sequence \(\displaystyle \{a_{n}\}\) be defined as \(\displaystyle a_{n} =5+\frac{1}{n}\). Find the value of \(\displaystyle \lim\limits _{n\rightarrow \infty } \ f( a_{n}) +g( a_{n})\).

Answer

\(40\)

Solution

We have:

\[ \begin{equation*} \lim\limits _{n\rightarrow \infty } \ f( a_{n}) =\lim\limits _{n\rightarrow \infty } \ f\left( 5+\frac{1}{n}\right) =8 \end{equation*} \]

And:

\[ \begin{equation*} \lim\limits _{n\rightarrow \infty } g( a_{n}) =\lim\limits _{n\rightarrow \infty } \ 2^{5+\frac{1}{n}} =2^{5} \end{equation*} \]

Summing the two limits, we get the required limit as \(\displaystyle 40\).