   Chapter 2.5, Problem 65E

Chapter
Section
Textbook Problem

Prove that cosine is a continuous function.

To determine

To show: The cosine function is continuous on a. That is, limh0cos(a+h)=cosa.

Explanation

From theorem 6, the statement is limh0sinθ=0 and limh0cosθ=1.

Proof:

Consider limh0cos(a+h).

Use the trigonometric identity cos(a+h)=cosacoshsinasinh as follows,

limh0cos(a+h)=limh0(cosacoshsinasinh)=limh0(cosacosh)limh0(sinasinh)=limh0cosalimh0coshlimh0sinalim</

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