Question

prove that lim sin x/x =1, (where x is in radian measure) and hence evaluate lim sin ax/bx

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Answer 1

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Answer 2

To Prove:

lim sin x/x =1, (where x is in radian measure)

To Find:

Find the value of

sin ax/bx

Step-by-step explanation:

1. The area of △ABC is 1/2sin(x). The area of the shaded wedge is 1/2x, and the area of △ABD is 1/2tan(x). By inclusion, we get

1/ 2tan(x) ≥ 1/2x ≥ 1/2sin(x)...............(1)

Dividing (1) by 1/2sin(x) and taking reciprocals, we get

cos(x) ≤ sin/x/x ≤1 ...............(2)

Since sinx/x and cos(x) are even functions, eq (2) is valid for any non-zero x between −π2 and π2. Furthermore, since cos(x) is continuous near 0 and cos(0)=1, we get that

limx→0 sinx/x=1 , hence proved.

2.  Sin(ax/bx)

Lim x→0 Sin(ax/bx) =

=Lim x→0(Sin x/x )(ax/bx)

=1.(a/b)

=a/b

Hence the value of Lim x→0 sin(ax/bx) is given as a/b.