About the computation of the limit $ \lim_{x \to \infty} \frac{1^{99} + 2^{99} + \cdots + x^{99}}{x^{100}} $

Question

I was reading this post and I don't understand why I can't do this:

\begin{align*} \lim_{x \to \infty} \frac{1^{99} + 2^{99} + \cdots + x^{99}}{x^{100}} &= \lim_{x \to \infty} \frac{1^{99}}{x^{100}} + \lim_{x \to \infty} \frac{2^{99}}{x^{100}} + \cdots + \lim_{x \to \infty} \frac{x^{99}}{x^{100}} \\ &= 0 + 0 + \cdots + \lim_{x \to \infty} \frac{1}{x} \\ &= 0 \end{align*}

I know that isn't the correct answer but I want to know why it's not. Why does it fail?

Answer 1

It fails because you're incorrectly using the linearity of limits. You can split a limit over a fixed sum, but the sum you're examining expands with $x$. Just like Riemann sum does (hint, hint!).