   Chapter 3.3, Problem 47E

Chapter
Section
Textbook Problem

Find the limit. lim θ → 0   cos θ − 1 2 θ 2

To determine

To find: The limit of limθ0cosθ12θ2.

Explanation

Limit law used:

Suppose that the limits limxaf(x) and limxag(x) exist, then

limxa(f(x)g(x))=limxaf(x)limxag(x)

Result used:

The limit of limx0sinxx=1.

Calculation:

Obtain limit of the function.

That is, compute limθ0cosθ12θ2.

Divide the numerator and the denominator by cosθ+1.

limθ0cosθ12θ2=limθ0(cosθ12θ2×cosθ+1cosθ+1)=limθ0(cos2θ12θ2(cosθ+1))            [(a2b2)=(a+b)(ab)]=limθ0sin2θ2θ2(cosθ+1)                [Qsin2θ+cos2θ=1]

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