Evaluating a Limit by Applying a Proven Result

Question

In a question I previously posted (Proving that $\lim\limits_{n \rightarrow \infty} \int^b_a f_n = \int^b_a f$) I (roughly) proved that: $\lim\limits_{n \rightarrow \infty} \int^b_a f_n = \int^b_a f$

But as an extension of this question I (believe) I need to apply it to work out: $$ \lim_{n \rightarrow \infty} \int^2_1 \sqrt{1+ \frac{e^x}{xn}}dx$$

Am I correct in saying that I need to apply $\lim\limits_{n \rightarrow \infty} \int^b_a f_n = \int^b_a f$ in order to solve this question or is a more standard approach required.

Answer 1

$1\lt (1+e^x/(xn))^{1/2} \lt $

$(1+e^2/n)^{1/2};$

$\displaystyle{\int_{1}^{2}}1dx \lt \int_{1}^{2}(1+e^x/(nx))^{1/2}dx \lt (1+e^2/n)^{1/2}\int_{1}^{2}1dx;$

Take the limit.

Used: Monotony of Riemann integral.