Limit of algrabic function

1. \( \displaystyle \lim_{x\to a}\frac{x^n-a^n}{x-a}=n{a^{n-1}}\)
   

   Solution
   We know that
   \( \frac{x^n-a^n}{x-a}=\frac{(x-a)(x^{n-1}+ax^{n-2}+a^2x^{n-3}+...+a^{n-1})}{x-a}\)
   or \( \frac{x^n-a^n}{x-a}=(x^{n-1}+ax^{n-2}+a^2x^{n-3}+...+a^{n-1})\)
   Thus, taking limit as \( x \to a\), we get
   \(\displaystyle \lim_{x\to a} \frac{x^n-a^n}{x-a}=\lim_{x\to a}(x^{n-1}+ax^{n-2}+a^2x^{n-3}+...+a^{n-1})\)
   or \(\displaystyle \lim_{x\to a} \frac{x^n-a^n}{x-a}=(a^{n-1}+a.a^{n-2}+a^2.a^{n-3}+...+a^{n-1})\)
   or \(\displaystyle \lim_{x\to a} \frac{x^n-a^n}{x-a}=n a^{n-1}\)
   This completes the proof