6.1.3 Limits Of Trigonometric Functions
| 6.1.3 |
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The following infinite limits can be visualized easily in Fig. 1.1.
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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Theorem 3.1
| 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.
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 [3.1] is no longer valid.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 sin2 x + cos2 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.
Remark 3.1
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| 4. The Fundamental Limit Applied To Sine Of Functions |
Example 4.1
Find:
Solution 1
EOS
Solution 2
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).
Example 4.2
Evaluate:
Solution
EOS
Example 4.3
Find:
Solution
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.
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1. Find each of the following limits if it exists.
Solution
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2. Find each of the following limits if it exists.
Solution
doesn’t exist.
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3. Find each of the following limits if it exists.
Solution
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4. Find each of the following limits if it exists.
Solution
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5. Find each of the following limits if it exists.
Solution
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