Science:Math Exam Resources/Courses/MATH110/April 2016/Question 04 (d)/Solution 1

Following the hint, as $x\to 5^{-}$, the denominator $(x-5{)}^2\to 0^{+}$ (note that the direction we approach 0 doesn't actually matter here since the denominator is squared!), while the numerator $x\to 5,$ so $\underset{x\to 5^{-}}{\lim }\frac{x}{(x-5{)}^2}=\mathrm{\infty }.$

Thus, the limit does not exist, and instead goes to $+\mathrm{\infty }.$