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 >> 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?

Solution 1:

Assume $a>b$ $$a^x\leq a^x+b^x\leq 2a^x$$

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

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

Solution 2:

Hint : If $a>b$, consider $a^x+b^x=a^x(1+(\frac{b}{a})^x)$ and use the fact that the expression in paranthesis tends to $1$.