How do you find the limit of $[4x+sinx] / [2x-sinx]$ as x approaches infinity?

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Answer 1 · 2018-04-05T09:47:46

Divide with $x$ denominator and numerator to get

$(4+sinx/x)/(2-sinx/x)$

Because the following holds

$|sinx|<=1$

Hence

$|sinx|/x<=1/x$

As $x->oo$ we have that $sinx/x->0$

Hence the requested limit is

$lim_(x->oo)(4+sinx/x)/(2-sin/x)=(4+0)/(2-0)=2$

How do you find the limit of [4x+sinx] / [2x-sinx][4x+sinx] / [2x-sinx] as x approaches infinity?