Find-lim-x-0-4x-1-4x-

Question Number 160658 by HongKing last updated on 04/Dec/21

Find : \lim_{x \to 0} 4 x [\frac{1}{4 x}] = ?

Commented by mr W last updated on 04/Dec/21

\underset{x \to 0}{\lim 4} x [\frac{1}{4 x}]

= \lim_{t \to \infty} \frac{[t]}{t} = L

t - 1 < [t] ⩽ t

\lim_{t \to \infty} \frac{t - 1}{t} < L ⩽ \lim_{t \to \infty} \frac{t}{t}

1 < L ⩽ 1

\Rightarrow L = 1

Commented by mnjuly1970 last updated on 04/Dec/21

v e r y n i c e s o l u t i o n s i r W

Commented by HongKing last updated on 04/Dec/21

Nice solution my dear Sir thank you

Answered by mathmax by abdo last updated on 04/Dec/21

we have [\frac{1}{4 x}] ⩽ \frac{1}{4 x} < [\frac{1}{4 x}] + 1 \Rightarrow \frac{1}{4 x} - 1 < [\frac{1}{4 x}] ⩽ \frac{1}{4 x} \Rightarrow

for x > 0 we have 1 - 4 x < 4 x . [\frac{1}{4 x}] ⩽ 1 \Rightarrow

\lim_{x \to 0^{+}} 4 x [\frac{1}{4 x}] = 1

Commented by HongKing last updated on 04/Dec/21

T h a n k y o u s o m u c h m y d e a r S i r v o o l