How to evaluate $ \lim_{x\rightarrow +\infty } \sqrt[x]{a^x+b^x} = ? $ If a>0 and b>0, 
$ \lim_{x\rightarrow +\infty } \sqrt[x]{a^x+b^x} = ? $

What I was trying to do:
Suppose a>b. Then, for sufficiently large values of x, $ a^x &gt;&gt; b^x $; so $\sqrt[x]{a^x+b^x} \rightarrow \sqrt[x]{a^x} \rightarrow a $ when $x \rightarrow +\infty$. 

Is that idea correct? How can I formalize it?
Assume $a&gt;b$
$$a^x\leq a^x+b^x\leq 2a^x$$

$$(a^x)^{\frac{1}{x}}\leq (a^x+b^x)^{\frac{1}{x}}\leq (2a^x)^{\frac{1}{x}}$$

$$a\leq (a^x+b^x)^{\frac{1}{x}}\leq 2^{\frac{1}{x}}a$$
You can go this way. Assuming $b&lt;a$


  $$ (a^x+b^x)^{1/x} = a \,e^{\frac{1}{x}\ln(1+(b/a)^x) }= a \,e^{\frac{1}{x}((b/a)^x-(b/a)^{2x}/2+\dots ) }\longrightarrow _{x\to \infty } a .$$


I think you can finish it!

Note: You need the following power series 


  $$ \ln(1+t) = t-\frac{t^2}{2}+ \frac{t^3}{3}-\dots\,.  $$