Question

Evaluate lim x→1 (x 8 -1 /x5 -1)​

Answer 1

$\tt \displaystyle \lim_{x \to 1} \tt \frac{{x}^{8} - 1}{{x}^{5} - 1 } \\ \\ \tt Dividing\:numerator\:and\:denominator\:by\:x-1 \\ \\ \tt \displaystyle \lim_{x \to 1} \tt \frac{\frac{{x}^{8} - 1}{x-1}}{\frac{{x}^{5} - 1 }{x-1}} \\ \\ \tt \boxed{\bold{\underline{\green{ \tt FORMULA: \displaystyle \lim_{x \to a } \tt \frac{{x}^{n} - {a}^{n}}{x - a} = n.{a}^{n-1} }}}}\\ \\ \tt \frac{ \displaystyle \lim_{x \to 1} \tt \frac{{x}^{8} - {1}^{8}}{x-1}}{ \displaystyle \lim_{x \to 1} \tt \frac{{x}^{5}-{1}^{5}}{x-1}} \\ \\ \tt \frac{8.{1}^{8-1}}{5.{1}^{5-1}} \\ \\ \tt \frac{8}{5} \\ \\ \tt \boxed{\bold{\underline{\red{ \tt \displaystyle \lim_{x \to 1} \tt \frac{{x}^{8} - 1}{{x}^{5} - 1 } = \frac{8}{5} }}}}$