find d limit as x -∞

Question

find d limit as x ->∞

{x +e^x +e^2x}^(1/x)

plz show step

Steve · Answer

(e^{^2x})^{^1/x} < (x+e^{^x}+e^{^2x})^{^1/x} < (2e^{^2x})^{^1/x}

e^² < (x+e^{^x}+e^{^2x})^{^1/x} < 2^{^1/x}e^²

Now take the limits, and since 2^{^1/x}-> 1,

e^² < (x+e^{^x}+e^{^2x})^{^1/x} < e^²

so

(x+e^{^x}+e^{^2x})^{^1/x} = e^²

Or, you can take the log of the limit, and then use l'Hospital's Rule