I've encountered some difficult problems in my textbook that I can't solve. Any help would be appreciated. This is not homework—I'm just looking to advance my knowledge, that's all.

Question 1: (SOLVED, but I still don't understand the general method for solving these kinds of problems)

For each $\displaystyle \epsilon > 0 $, find a $\displaystyle \omega$ so that $\displaystyle x > \omega \implies \left| {\frac{\sqrt{x^2 + 1}}{x} - 1 }\right| < \epsilon$ is true. What does this show?

(By the way, how do I get the absolute characters to extend all the way in LaTeX?)SOLVED: Use \left| and \right|

Question 2: (SOLVED, but I still don't understand the general method for solving these kinds of problems)

Show that $\displaystyle x^3 \to 1$ when $\displaystyle x \to 1$ using the definition of limits.

This is what I've got so far:

As soon as we've chosen a $\displaystyle \epsilon > 0$, then we'll be able to find a $\displaystyle \delta > 0$ so that

$\displaystyle |x^3 - 1| < \epsilon$ when $\displaystyle 0 < |x - 1| < \delta$.

Note:

$\displaystyle |x^3 - 1| = |x - 1| \cdot |x^2 + x + 1|$

Question 3: (SOLVED)

Is $\displaystyle f(x) = \lim_{n \to \infty} \frac{x^n}{1 + x^n}$ continuous for all $\displaystyle x \geq 0$?

Is $\displaystyle f(x) = \lim_{n \to \infty} \frac{x + x^2 + ... + x^n}{1 + x + ... + x^n}$ continuous for all $\displaystyle x \geq 0$?