Calculus Of One Real Variable – By Pheng Kim Ving

6.1.3 
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1. Infinite Limits 
The following infinite limits can be visualized easily in Fig. 1.1.
Fig. 1.1
sin
x = v,

Of course these limits can be proved
by using the definitions of the functions in terms of the sine and cosine
functions. For
example:
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2. No Limits At Infinity 
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3. A Fundamental Limit 
Fig. 3.1
As x approaches
0, (sin x)/x appears to approach 1.


Fig. 3.2
As x approaches
0, (sin x)/x appears to approach 1.

Let x be an angle measured in radians. Then: 
Proof
Suppose x is small and positive. Refer to
Fig. 3.1. Clearly:
(area of triangle OPU) < (area of circular sector OPA) < (area of triangle OZA).
Now suppose x is small and negative. Let t = – x > 0. Then, utilizing the identity sin (– x) = – sin x we obtain:
EOP
The Necessity Of The Radian Measure
In the above proof, the hypothesis that x
is measured in radians is used to get to the fact that the value of the area of
the
circular sector OPA is (1/2)x. What happens if x
isn't in radians?
Let's do some numerical calculations, as displayed in
Fig. 3.3, to see what happens if x is measured
in degrees.

Fig. 3.3
As x^{o} approaches 0, (sin x^{o})/x^{o}
doesn't appear

OK. Now what happens if x isn't in radians
is that Eq. [3.1] is no longer valid. Hence, the hypothesis of Theorem 3.1,
that
the angle x is measured in radians, is indeed
necessary for the conclusion, Eq. [3.1], to be valid.
Remark that trigonometric identities such as sin^{2} x + cos^{2} x = 1 or sin
(x + y)
= sin
x cos y + cos
x sin y don't require
that the angle x is in radians. For such
identities, the unit of measurement for x may be the
degree as well as the radian.
Establishing them didn't require that they were in any particular unit.
It's A Fundamental Limit
The limit in Eq. [3.1] is classified as a fundamental trigonometric limit. The reason is that
it's, well, fundamental, or basic,
in the development of the calculus for trigonometric functions. As we'll see,
the derivatives of trigonometric functions,
among other things, are obtained by using this limit.
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4. The Fundamental Limit Applied To Sine Of Functions 
Example 4.1
Find:
EOS
EOS
General Case
In general we can use this rule:

To use this rule, the argument of sin and the
denominator must be the same function. In the above statement of the rule
the argument of sin and the denominator are the function f(x).
Evaluate:
EOS
Find:
EOS
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5. Limit Of Quotients Involving Cosine 
So:

Remarks 5.1
i. Since we
use Eq. [3.1] to derive Eq. [5.1], the angle x in Eq.
[5.1] must be measured in radians in order for that
equation to be valid.
Problems & Solutions 
1. Find each of the following limits if it exists.
Solution
2. Find each of the following limits if it exists.
Solution
doesn't exist.
3. Find each of the following limits if it exists.
Solution
4. Find each of the following limits if it exists.
Solution
5. Find each of the following limits if it exists.
Solution
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