Question

Evaluate the following limits: ${\displaystyle \underset{x\to 5}{lim}\left[\frac{x^3-125}{x^2-25}\right]}$

Answer 1

${\displaystyle \underset{x\to 5}{lim}\frac{x^3-125}{x^2-25}}$

= ${\displaystyle \underset{x\to 5}{lim}\frac{\frac{x^3-125}{x^2-25}}{\frac{x^2-25}{x-5}} ...\left[\begin{array}{c}\text{As} x\to 5\text{,} x\ne 5\\ \therefore x-5\ne 0\\ \text{Divide Numerator and}\\ \text{Denominator by} x-5.\end{array}\right]}$

= ${\displaystyle \underset{x\to 5}{lim}\frac{\left(\frac{x^3-5^3}{x-5}\right)}{\left(\frac{x^2-5^2}{x-5}\right)}}$

= ${\displaystyle \frac{3{\left(5\right)}^2}{2{\left(5\right)}^1} ...[\underset{x\to \text{a}}{lim}\frac{x^{\text{n}}-{\text{a}}^{\text{n}}}{x-\text{a}}={\text{na}}^{\text{n}-1}]}$

= ${\displaystyle \frac{15}{2}}$