Question

Construct the graph of the function $y=f\left(x\right)$, where $f\left(x\right)=\underset{n\to \infty }{lim}\frac{x^n-x^{-n}}{x^n+x^{-n}}$, $x>0$

Answer 1

$f\left(x\right)=\underset{n\to \infty }{lim}\frac{x^n-x^{-n}}{x^n+x^{-n}}$

$\Rightarrow f\left(x\right)=\underset{n\to \infty }{lim}\frac{x^n-\frac{1}{x^n}}{x^n+\frac{1}{x^n}}$

$\Rightarrow f\left(x\right)=\underset{n\to \infty }{lim}\frac{x^n(1-\frac{1}{x^{2n}})}{x^n(1+\frac{1}{x^{2n}})}$

$\Rightarrow f\left(x\right)=\underset{n\to \infty }{lim}\left(\frac{1-\frac{1}{x^{2n}}}{1+\frac{1}{x^{2n}}}\right)$

$\Rightarrow f\left(x\right)=\frac{1-0}{1+0}$

$\Rightarrow f\left(x\right)=1$

So, $y=f\left(x\right)=1$

Now the graph is as (Image)