Limit of function

[4c0]

How to find limit of function \(\displaystyle xe^{\frac{x}{2\left ( x+2 \right )}}-xe^{1/2}\) as x->inf?
Is there a way to do this without shift x=1/t?

 

[lookagain]

How to find limit of function \(\displaystyle xe^{\frac{x}{2\left ( x+2 \right )}}-xe^{1/2}\) as x->inf?
Is there a way to do this without shift x=1/t?

Expand each e raised to each of those exponents \(\displaystyle e^u = 1 + u + \dfrac{u^2}{2!} + \dfrac{u^3}{3!} + ...\)


First, let u = \(\displaystyle \frac{x}{x + 2} \ \ and \ \ then \ let \ u = \frac{1}{2}.\)


After subtracting corresponding pairs, multiply each difference by x.


So far I am getting \(\displaystyle -1 - \frac{1}{2} - \frac{1}{8} - \frac{1}{48} - ...\)