Question

$\underset{x\to 5}{\lim }\frac{x^3-125}{x^2-7x+10}$

Answer 1

$$\begin{gathered} \underset{x\to 5}{\lim }\left[\frac{x^3-125}{x^2-7x+10}\right] \\ \mathrm{It}\mathrm{is}\mathrm{of}\mathrm{the}\mathrm{form}\frac{0}{0}. \\ \underset{x\to 5}{\lim }\left[\frac{x^3-5^3}{x^2-2x-5x+10}\right] \\ =\underset{x\to 5}{\lim }\left[\frac{\left(x-5\right)\left(x^2+5x+5^2\right)}{x\left(x-2\right)-5\left(x-2\right)}\right]\left[\because A^3-B^3=\left(A-B\right)\left(A^2+AB+B^2\right)\right] \\ =\underset{x\to 5}{\lim }\left[\frac{\left(x-5\right)\left(x^2+5x+25\right)}{\left(x-2\right)\left(x-5\right)}\right] \\ =\frac{5^2+5\times 5+25}{5-2} \\ =\frac{75}{3} \\ =25 \end{gathered}$$