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

Answer 1

Luke Alvarez

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#

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Answer 2

Isabella Ackerman

To find the limit of [4x+sinx] / [2x-sinx] as x approaches infinity, we can use the concept of dominant terms. By dividing both the numerator and denominator by x, we get [4 + (sinx/x)] / [2 - (sinx/x)]. As x approaches infinity, sinx/x approaches 0. Therefore, the limit simplifies to 4/2, which is equal to 2.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.