solving the limit of $(e^{2x}+x)^{1/x}$ I tried to solve for the following limit: 
$$\lim_{x\rightarrow \infty} (e^{2x}+x)^{1/x}$$
and I reached to the indeterminate form: 
$${4e^{2x}}\over {4e^{2x}}$$ 
if I plug in, I will get another indeterminate form! 

Another way, considering $$A=(e^{2x}+x)^{1/x}$$ Taking logarithms $$\log(A)=\frac 1x \log(e^{2x}+x)=\frac 1x \left(\log(e^{2x})+\log(1+\frac x {e^{2x}})\right)=\frac 1x \left(2x+\log(1+\frac x {e^{2x}})\right)$$ $$\log(A)=2+\frac 1x \log(1+\frac x {e^{2x}})\approx 2+\frac 1x \frac x {e^{2x}}=2+\frac 1 {e^{2x}}$$
