lim-x-3-x-3-x-3-x-3-x-

Question Number 137737 by bemath last updated on 06/Apr/21

\lim_{x \to - \infty} \frac{3^{x} - 3^{- x}}{3^{x} + 3^{- x}} = ?

Answered by greg_ed last updated on 06/Apr/21

L^{'} {Hopital}^{'} s rule

\lim_{x \to - \infty} \frac{3^{x} - 3^{- x}}{3^{x} + 3^{- x}} = \lim_{x \to - \infty} \frac{\frac{d}{d x} (3^{x} - 3^{- x})}{\frac{d}{d x} (3^{x} + 3^{- x})}

= \lim_{x \to - \infty} \frac{\frac{l n (3) \times 9^{x} + l n (3)}{3^{x}}}{\frac{l n (3) \times 9^{x} - l n (3)}{3^{x}}}

= \lim_{x \to - \infty} \frac{l n (3) \times 9^{x} + l n (3)}{l n (3) \times 9^{x} - l n (3)}

= \lim_{x \to - \infty} \frac{9^{x} + 1}{9^{x} - 1}

since \lim_{x \to - \infty} 9^{x} = 0, then :

\lim_{x \to - \infty} \frac{3^{x} - 3^{- x}}{3^{x} + 3^{- x}} = - 1 .

Answered by bemath last updated on 06/Apr/21

Facebook Tweet Pin