TITLE: $\lim_{x \to +\infty}\frac{x+\sqrt{x}}{x-\sqrt{x}}$
QUESTION [6 upvotes]: Calculate:
$$\lim_{x \to +\infty}\frac{x+\sqrt{x}}{x-\sqrt{x}}$$
i tried to take $X=\sqrt{x}$ we give us 
when $x \to 0$ we have $X \to 0$
But i really don't know if it's a good idea

REPLY [3 votes]: You may have noticed the similarity between the nominator and the denominator. If you choose to make use of this you can express the ratio in the following way. $$\frac{x+\sqrt{x}}{x-\sqrt{x}}=\frac{(x-\sqrt{x})+2\sqrt{x}}{x-\sqrt{x}} = 1+\frac{2}{\sqrt{x}-1}\ .$$
If $x$ is a large positive number, then so will its square-root be a large positive number; consequently the original ratio will be close to $1$. If you like, you can formalize this using an $\epsilon$-$\delta$-argument, but I will leave that to you.