Chapter 2 | Limits
173
Figure 2.30 The sine and tangent functions are shown as lines on the unit circle.
By dividing by sin θ in all parts of the inequality, we obtain 1< θ sin θ
< 1 cos θ .
Equivalently, we have
1> sin θ
θ >cos
θ .
sin θ
Since lim
1=1= lim
cos θ , we conclude that lim θ →0 +
θ =1.
By applying a manipulation similar to that used
θ →0 +
θ →0 +
sin θ
in demonstrating that lim θ →0 −
sin θ =0, we can show that lim θ →0 −
θ =1.
Thus,
sin θ
(2.18)
lim θ →0
θ =1.
1−cos θ θ
In Example 2.25 we use this limit to establish lim θ →0
=0.
This limit also proves useful in later chapters.
Example 2.25 Evaluating an Important Trigonometric Limit
1−cos θ θ .
Evaluate lim θ →0
Solution In the first step, we multiply by the conjugate so that we can use a trigonometric identity to convert the cosine in the numerator to a sine:
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