Calculus Of One Real Variable By Pheng Kim Ving
Chapter 6: The Trigonometric Functions And Their Inverses Section 6.1.3: Limits Of Trigonometric Functions

 

6.1.3
Limits Of Trigonometric Functions

 

 

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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,
cos x = u,
tan x = z,
cot x = w,
sec x = q,
csc x = r.

 

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.

 

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.

 

Fig. 3.3

 

As xo approaches 0, (sin xo)/xo doesn't appear
to approach 1.

 

 

 

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 [3.1] is no longer valid.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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Problems & Solutions

 

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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