lim-x-0-x-2-tanxsinx-grestest-integer-function-

Question Number 60410 by tanmay last updated on 20/May/19

\lim_{x \to 0} [\frac{x^{2}}{t a n x s i n x}] [.] = g r e s t e s t i n t e g e r f u n c t i o n

Commented by kaivan.ahmadi last updated on 20/May/19

i f 0 < x < \frac{π}{2}

s i n x < x < t g x \Rightarrow x s i n x < x^{2} < x t g x \Rightarrow

\frac{x s i n x}{s i n x t g x} < \frac{x^{2}}{s i n x t g x} < \frac{x t g x}{s i n x t g x} \Rightarrow

\frac{x}{t g x} < \frac{x^{2}}{s i n x t g x} < \frac{x}{s i n x} \Rightarrow

[\frac{x}{t g x}] < [\frac{x^{2}}{s i n x t g x}] < [\frac{x}{s i n x}] \Rightarrow

0 < [\frac{x^{2}}{s i n x t g x}] < 1 \Rightarrow

{l i m}_{x \to 0} [\frac{x^{2}}{s i n x t g x}] = 0

Commented by tanmay last updated on 20/May/19

e x c l l e n t

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