Why can there be an infinite difference between two functions as x grows large, but a ratio of 1? I learned in grade school that the closer $a$ and $b$ are to one another, the closer $\frac{a}{b}$ is going to be to $1$. For example, $\frac{3}{\pi}$ is pretty close to 1, and $\frac{10^{100}}{42}$ isn't even close to 1.
So, why is:
$$\lim_{x\to\infty} \frac{x^{2}}{x^{2}+x} = 1$$
But:
$$\lim_{x\to\infty}[(x^2+x)-(x^2)] = \infty$$ ?
Seems pretty counterintuitive. What's going on here?

The reason is that $$\frac{x^2}{x^2+x}=\frac{x^2}{x^2+x}\frac{1/x^2}{1/x^2}=\frac{1}{1+1/x}$$
The fraction is always less than $1$ (for positive $x$), as the denominator is bigger than $1$.  However, as $x\to\infty$, we have $\frac{1}{x}\to 0$, so the fraction gets closer and closer to $1$.
