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Tuesday, April 28, 2015

Calculate limx(sinx+1sinx)

Calculate limx(sinx+1sinx).

Let:

L=limx(sin(x+1)sin(x))

Application of the sum-to-product identity:

sin(α)sin(β)=2sin(αβ2)cos(α+β2)

And the limit property:

limxc(kf(x))=klimxc(f(x))

Allows us to write:

L=2limx(sin(x+1x2)cos(x+1+x2))

Rationalization of the numerator of the sine function gives us:

L=2limx(sin(12(x+1+x))cos(x+1+x2))

Now, using the property of limits:

limxc(f(x)g(x))=(limxc(f(x)))(limxc(g(x)))

We obtain:

L=2limx(sin(12(x+1+x)))limx(cos(x+1+x2))

The first limit goes to zero, and the second limit is bounded, hence:

L=0

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