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 x^o approaches 0, (sin x^o)/x^o 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 [3.1] is no longer valid.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).