Science:Math Exam Resources/Courses/MATH100/December 2011/Question 01 (a)/Solution 1

When we evaluate the function at $t=-2$ we obtain 0/0. Since this is a rational function, it means that we can factor $t+2$ in each of the denominator and numerator:

$\displaystyle 2t^{2}+t-6=(t+2)(2t-3)$

and

$\displaystyle t^{2}-4=(t+2)(t-2)$

Hence

${\begin{aligned}\lim _{t\to -2}{\frac {2t^{2}+t-6}{t^{2}-4}}&=\lim _{t\to -2}{\frac {(t+2)(2t-3)}{(t+2)(t-2)}}\\&=\lim _{t\to -2}{\frac {(2t-3)}{(t-2)}}\\&={\frac {2(-2)-3}{(-2)-2}}\\&={\frac {7}{4}}\end{aligned}}$