Consider the function y x2 x2 1 Find the critical values v

Consider the function y = x2 / x2 -1 Find the critical values, vertical and horizontal asymptotes of f. Are any of the critical values extrema? If so, state whether they are local max and local min.

Solution

f(x) = x²/(x² - 1) = x²/(x-1)(x+1)

= 1 + 1/(x² - 1)

f \' (x) = - 2x/(x² - 1)²

(a) For critical points, f\'(x) = 0

- 2x/(x² - 1)² = 0

x = 0 is the critical point

(b) For vertical asymptotes, find x such that f(x) -> infinity

x = ± 1 are the vertical asymptotes

(c) For horizonatl asymptotes, find y such that x-> infinity

y = x²/(x² - 1)

y(x² - 1) = x²

x² = y/(y-1)

We see that as y -> 1, x-> infinity

y = 1 is a horizontal asymptote