$$\begin{align} \lim\limits_{x \to 1} \frac{x^2 - 4x + 3}{x^2 -3x + 2} &= \lim\limits_{x \to 1} \frac{(x-3)(x-1)}{(x-2)(x-1)} \\[2ex] &= \lim\limits_{x \to 1} \frac{x-3}{x-2} \\[2ex] &= \frac{1-3}{1-2} \\[2ex] &= \frac{-2}{-1} \\[2ex] &= 2 \end{align}$$

$$\begin{align} \lim\limits_{x \to 1} \frac{x - 1}{\sqrt{x} - 1} &= \lim\limits_{x \to 1} \frac{x - 1}{\sqrt{x} - 1}\cdot \frac{\sqrt{x}+1}{\sqrt{x}+1} \\[2ex] &= \lim\limits_{x \to 1} \frac{(x-1)(\sqrt{x}+1)}{x-1} \\[2ex] &= \lim\limits_{x \to 1} (\sqrt{x}+1) \\[2ex] &= \sqrt{1}+1 \\[2ex] &= 2 \\ \end{align}$$

$$\begin{align} \lim\limits_{x \to \infty} \frac{2x^2 +4x - 1}{x^2 - 3x - 4} &= \lim\limits_{x \to \infty} \frac{(2x^2 +4x - 1)\cdot \frac{1}{x^2}}{(x^2 - 3x - 4) \cdot \frac{1}{x^2}} \\[2ex] &= \lim\limits_{x \to \infty} \frac{(\frac{2x^2}{x^2} +\frac{4x}{x^2} - \frac{1}{x^2})}{(\frac{x^2}{x^2} - \frac{3x}{x^2} - \frac{4}{x^2})} \\[2ex] &= \lim\limits_{x \to \infty} \frac{(2 + \frac{4}{x} - \frac{1}{x^2})}{(1 - \frac{3}{x} - \frac{4}{x^2})} \\[2ex] &= \frac{(2 + \lim\limits_{x \to \infty} \frac{4}{x} - \lim\limits_{x \to \infty} \frac{1}{x^2})}{(1 - \lim\limits_{x \to \infty} \frac{3}{x} - \lim\limits_{x \to \infty} \frac{4}{x^2})} \\[2ex] &= \frac{2}{1} \\[2ex] &= 2 \\ \end{align}$$