Question

Draw the graph of the function `y= f(x) = lim_(n to oo) (x^(2n)-1)/(x^(2n)+1)`. Is this function same as the function `g(x) = "sgn"(x^(2)-1)`.

Answer 1

`f(x) =underset(ntooo) (lim) ((x^(2))^(n)-1)/((x^(2))^(n)+1)`
`" "=underset(n to oo) (lim) (1-(1)/((x^(2))^(n)))/(1+(1)/((x^(2))^(n)))`
`" "={{:(-1",",,0le x^(2)lt1),(0",",,x^(2) =1),(1",",, x^(2) gt 1):}`
`" "={{:(1",",,xlt-1),(0",",,x=-1),(-1",",,-1 lt xlt 1), (0",",,x=1), (1",",,x gt 1):}`
Clearly, the above function is same as `g(x)= "sgn"(x^(2) -1)`.