A model for getting limitation

Method

If anything goes well, it indeed has a general model for you guys getting limitation from a limit equation.

if an equation is a fraction:
    if fraction + or - a fraction:
        reduce fractions into one fraction by setting a common denominator
    if the equation has a radical sign:
        get rid of it by rationalizing.
    if put giving_x into equation getting "0/0":
        do factorization, get rid of one factor which makes equation looks like "0/0"
if the equation is not a fraction: # combined with simple +-x/
    put giving_x in equation directly

Example

$$\begin{align*} & \lim_{x \to 1}{\frac{x^2 + 2x - 3}{x - 1}} \\ \\ = & \lim_{x \to 1}{\frac{(x + 3)(x - 1)}{x - 1}} \\ \\ = & \lim_{x \to 1}{\frac{x + 3}{1}} \\ \\ = & 1 + 3 \\ \\ = & 4 \end{align*}$$

$$\begin{align*} & \lim_{x \to +\infty}{\sqrt{x^2 - 1} - x} \\ \\ = & \lim_{x \to +\infty}{\frac{\sqrt{x^2 - 1} - x}{1}} \\ \\ = & \lim_{x \to +\infty}{\frac{(\sqrt{x^2 - 1} - x) (\sqrt{x^2 - 1} + x)}{1(\sqrt{x^2 - 1} + x)}} \\ \\ = & \lim_{x \to +\infty}{\frac{x^2 - 1 - x^2}{\sqrt{x^2 - 1} + x}} \\ \\ = & \lim_{x \to +\infty}{\frac{-1}{\sqrt{x^2 - 1} + x}} \\ \\ = & \lim_{x \to +\infty}{\frac{-1}{\frac{\sqrt{x^2 - 1} + x}{x}}} \\ \\ = & \lim_{x \to +\infty}{\frac{-1}{\sqrt{\frac{x^2}{x^2} - \frac{1}{x^2}} + \frac{x}{x}}} \\ \\ = & \lim_{x \to +\infty}{\frac{-1}{\sqrt{1 - \frac{1}{x^2}} + 1}} \\ \\ = & \frac{-1}{\sqrt{1 - \frac{1}{+\infty}} + 1} \\ \\ = & \frac{-1}{\sqrt{1 - 0} + 1} \\ \\ = & -\frac{1}{2} \end{align*}$$