How to prove this $\lim$ Let $n$ be integer, $n\geq 2$, and $P(x)$ be a polynomial with degree < $n-1$.
How to prove that $$\lim_{x\to\infty}\left(\sqrt[n]{(x+a)^n+P(x)}-x-a\right)=0$$
I try to use conjugate but it is very difficult.
Thank you for helping.  

Let $$A = \sqrt[n]{(x + a)^{n} + P(x)}, B = (x + a), C = \frac{P(x)}{(x + a)^{n}}, D = \frac{P(x)}{(x + a)^{n - 1}}$$ and then we can see that \begin{align}
A - B &= \frac{A^{n} - B^{n}}{A^{n - 1} + A^{n - 2}B + \dots + AB^{n - 2} + B^{n - 1}}\notag\\
&= \dfrac{P(x)}{(x + a)^{n - 1}\left\{\left(1 + \dfrac{P(x)}{(x + a)^{n}}\right)^{(n - 1)/n} + \left(1 + \dfrac{P(x)}{(x + a)^{n}}\right)^{(n - 2)/n} + \cdots + 1\right\}}\notag\\
&= \frac{D}{(1 + C)^{(n - 1)/n} + (1 + C)^{(n - 2)/n} + \cdots + 1}\tag{1}
\end{align} Note that $P(x)$ is a polynomial of degree less than $(n - 1)$ and hence $C, D$ both tend to $0$ as $x \to \infty$. Clearly in the equation $(1)$ we can see that the numerator tends to $0$ and denominator tends to $1 + 1 + \dots + 1 = n$ so that $(A - B) \to 0$ as $x \to \infty$.
