Evaluate lim(x→1)x^m - 1/x^n-1.

Question

Evaluate :

\(\lim\limits_{x \to 1}\frac{x^m-1}{x^n-1}\)

Answer 1

\(\lim\limits_{x \to 1}\frac{x^m-1}{x^n-1}\)

= \(\lim\limits_{x \to 1}\frac{x^m-1}{x-1}\times\frac{x-1}{x^n-1}\)

= \(\frac{m.1^{m-1}}{n.1^{n-1}}\)

[∵ \(\lim\limits_{x \to a}\frac{x^n-a^n}{x-a}=na^{n-1}\)]

=\(\frac{m.1}{n.1}\)

= \(\frac{m}{n}\)

Answer 2

Apply L-H rule by differentiating numerator and denominator simultaneously.

Iim x-->1(X^m -1)/(xⁿ-1)

mx^m-1/nx^n-1

^m/n