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

We can be a bit clever and factor $x^{2}-3x-4=(x-4)(x+1)$ 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

${\begin{aligned}\lim _{x\rightarrow 4}{\frac {x^{2}-3x-4}{{\sqrt {x}}-2}}&=\lim _{x\rightarrow 4}{\frac {(x-4)(x+1)}{{\sqrt {x}}-2}}\\&=\lim _{x\rightarrow 4}{\frac {({\sqrt {x}}-2)({\sqrt {x}}+2)(x+1)}{{\sqrt {x}}-2}}\\&=\lim _{x\rightarrow 4}({\sqrt {x}}+2)(x+1)\\&=({\sqrt {4}}+2)(4+1)\\&=20\end{aligned}}$