   Chapter 2.4, Problem 47E

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

To determine

To find: the limit of the function.

Explanation

1) Formula:

i.

a2-b2=a-ba+b

ii.

limθ0sinθθ=1

2) Given:

limθ0cosθ-12θ2

3) Calculation:

limθ0cosθ-12θ2

To multiply top and bottom by cosθ+1

limθ0cosθ-12θ2=limθ0cosθ-12θ2cosθ+1cosθ+1

By formula a2-b2=a-ba+b then

limθ0cosθ-12θ2=limθ0cosθ2-122θ21cosθ+1

limθ0cosθ-12θ2=limθ0cos2θ-12θ21cosθ+1

By formula cos2θ-1=-sin2θ  then

limθ0cosθ-12θ2=limθ0-sin2θ2θ21cosθ+1

limθ0cosθ-12θ2=limθ0-sinθ22θ21cosθ+1

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