Science:Math Exam Resources/Courses/MATH104/December 2011/Question 01 (a)/Solution 3

We can be a bit clever and factor $x^{2} - 3 x - 4 = (x - 4) (x + 1)$ then factor $x - 4$ as a difference of squares where we think of $x$ as the square of $\sqrt{x}$ .

Hence

x - 4 = (\sqrt{x} - 2) (\sqrt{x} + 2)

.

Combining gives

\begin{aligned} \lim_{x \to 4} \frac{x^{2} - 3 x - 4}{\sqrt{x} - 2} & = \lim_{x \to 4} \frac{(x - 4) (x + 1)}{\sqrt{x} - 2} \\ = \lim_{x \to 4} \frac{(\sqrt{x} - 2) (\sqrt{x} + 2) (x + 1)}{\sqrt{x} - 2} \\ = \lim_{x \to 4} (\sqrt{x} + 2) (x + 1) \\ = (\sqrt{4} + 2) (4 + 1) \\ = 20 \end{aligned}