Calculate-limits-of-f-at-1-and-1-f-x-2x-3-x-1-x-2-

Question Number 94849 by mathocean1 last updated on 21/May/20

Calculate limits of f at + \infty; - 1 and 1

f (x) = 2 x + 3 - \frac{x}{1 - x^{2}}

Answered by mathmax by abdo last updated on 21/May/20

f (x) = 2 x + 3 + \frac{x}{x^{2} - 1} \Rightarrow \lim_{x \to + \infty} f (x) = \lim_{x \to + \infty} (2 x + 3 + \frac{1}{x})

= + \infty

we have f (x) = \frac{(2 x + 3) (1 - x^{2}) - x}{1 - x^{2}} = \frac{2 x - {2 x}^{3} + 3 - {3 x}^{2} - x}{1 - x^{2}}

= \frac{- {2 x}^{3} - {3 x}^{2} + x + 3}{1 - x^{2}} = \frac{{2 x}^{3} + {3 x}^{2} - x - 3}{(x - 1) (x + 1)} \Rightarrow

\lim_{x \to 1^{+}} f (x) = \frac{1}{0^{+}} = + \infty and \lim_{x \to 1^{-}} f (x) = \frac{1}{0^{-}} = - \infty

\lim_{x \to - 1^{+}} f (x) = \frac{- 1}{(- 2) \times o^{+}} = + \infty

\lim_{x \to - 1^{-}} f (x) = - \infty