Follow up question to find $\lim \limits_{k \to \infty} \int_{0}^{k} \left(1-\frac{x}{k}\right)^k \cdot e^{\frac{x}{3}} dx$ .

Question

I need to find

$$I=\lim_{k\to\infty}\left(\sum _{n=0}^kn!\ ^kC_n \left(\frac{3}{k}\right)^{n}-\dfrac{3^kk!e^{\tfrac k3}}{k^k} \right).$$

This was because I got stuck during an attempt to solve this question by methods accessible to a high-schooler. This is basically the error in the approximation of the right hand term by the sum, by this post. Any help is appreciated!

Answer is supposed to be

$$\dfrac 12$$

P.S. : If you want to see “my effort” then see my answer to the linked post. :)