Question

In the equation lim x- 0 sin 4x/2x I get that you multiply it by 2 to get 2sin4x/4x but how do you get an answer of 2?

Answer 1

`We\ know:\lim\limits_(x\to0)(\sin x)/(x)=1\\-------------------\\\lim\limits_(x\to0)(\sin4x)/(2x)=\lim\limits_(x\to0)(2\sin4x)/(2\cdot2x)=\lim\limits_(x\to0)2\cdot(\sin4x)/(4x)=2\lim\limits_(x\to0)(\sin4x)/(4x)=(*)\\\\subtitute\ 4x=t;\ x\to0\ then\ 4x\to0\ then\ t\to 0\\\\(*)=2\lim\limits_(t\to0)(\sin t)/(t)=2\cdot1=2`