Question

The value of $\underset{x\to 1}{lim}\frac{100}{1-x^{100}}-\frac{50}{1-x^{50}}$ is

Answer 1

$L=\underset{x\to 1}{lim}\frac{100}{1-x^{100}}-\frac{50}{1-x^{50}}$

Putting $x=\frac{1}{y}$, we get
$L=\underset{y\to 1}{lim}\frac{100y^{100}}{y^{100}-1}-\frac{50y^{50}}{y^{50}-1}\Rightarrow L=\underset{y\to 1}{lim}\frac{100(y^{100}-1+1)}{y^{100}-1}-\frac{50(y^{50}-1+1)}{y^{50}-1}\Rightarrow L=\underset{y\to 1}{lim}[100-50-(\frac{100}{1-y^{100}}-\frac{50}{1-y^{50}})]\Rightarrow L=50-L\therefore L=25$