calculate-lim-x-1-x-tan-pix-2x-3-

Question Number 40092 by maxmathsup by imad last updated on 15/Jul/18

c a l c u l a t e {l i m}_{x \to + \infty} \frac{1}{x} t a n (\frac{π x}{2 x + 3})

Commented by math khazana by abdo last updated on 26/Jul/18

w e h a v e \frac{π x}{2 x + 3} = \frac{π x}{2 x (1 + \frac{3}{2 x})} = \frac{π}{2} . \frac{1}{1 + \frac{3}{2 x}}

\sim \frac{π}{2} (1 - \frac{3}{2 x} + 0 (\frac{1}{x})) (x \to + \infty) \Rightarrow

t a n (\frac{π x}{2 x + 3}) \sim t a n (\frac{π}{2} - \frac{3 π}{4 x}) \sim \frac{1}{t a n (\frac{3 π}{4 x})} \Rightarrow

\frac{1}{x} t a n (\frac{π x}{2 x + 3}) \sim \frac{1}{x t a n (\frac{3 π}{4 x})} c h a n g e m e n t

\frac{3 π}{4 x} = t g i v e 4 t x = 3 π \Rightarrow {l i m}_{x \to + \infty} \frac{1}{x t a n (\frac{3 π}{4 x})}

= {l i m}_{t \to 0} \frac{1}{\frac{3 π}{4 t} t a n (t)} = \frac{4}{3 π} {l i m}_{t \to 0} \frac{1}{\frac{t a n t}{t}} = \frac{4}{3 π}

b e c a u s e {l i m}_{t \to 0} \frac{t a n t}{t} = 1