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The Limit Laws: Learn It 4

The Squeeze Theorem Cont.

We now use the squeeze theorem to tackle several very important limits. Although this discussion is somewhat lengthy, these limits prove invaluable for the development of the material in both the next section and the next module. The first of these limits is limθ0sinθ.

Evaluating the Limit of Sine as Theta Approaches Zero

Consider the behavior of sin(θ) as θ approaches zero. On the unit circle, sin(θ) corresponds to the y-coordinate, which also represents the arc’s height for a given angle, θ.

As θ gets closer to zero, particularly for 0<θ<π2, sin(θ) becomes smaller and approaches the angle’s measure itself, meaning sin(θ) is squeezed between 0 and θ .

A diagram of the unit circle in the x,y plane – it is a circle with radius 1 and center at the origin. A specific point (cos(theta), sin(theta)) is labeled in quadrant 1 on the edge of the circle. This point is one vertex of a right triangle inside the circle, with other vertices at the origin and (cos(theta), 0). As such, the lengths of the sides are cos(theta) for the base and sin(theta) for the height, where theta is the angle created by the hypotenuse and base. The radian measure of angle theta is the length of the arc it subtends on the unit circle. The diagram shows that for 0 < theta < pi/2, 0 < sin(theta) < theta.
Figure 7. The sine function is shown as a line on the unit circle.

First, consider the established inequalities for sin(θ) when θ  is between 0 and π2:

0<θ<π20<sinθ<θ

Now, as θ approaches zero from the positive direction, we know that sin(θ) also approaches zero because it is sandwiched between 0 and θ.

Mathematically, this can be expressed as:

limθ0+0=0 and limθ0+θ=0,

which, according to the Squeeze Theorem, compels sin(θ) to satisfy:

limθ0+sinθ=0.

The same principle applies when approaching zero from the negative side, where sin(θ) is negative but greater than θ:

π2<θ<0θ<sinθ<0
Here too, as θ approaches zero, sin(θ)  is “squeezed” to zero:
 
limθ00=0 and limθ0(θ)=0,
leading to the conclusion that:
 
limθ0sinθ=0.
Therefore, we can definitively state that the limit of sin(θ) as θ approaches zero from either direction is 0.

the limit of sin(θ)

limθ0sinθ=0

Evaluating the Limit of Cosine as Theta Approaches Zero

To evaluate the limit of cos(θ) as θ approaches zero, we rely on the fundamental Pythagorean identity which states that for any angle θ, the square of the cosine of θ plus the square of the sine of θ equals one:

cos2(θ)+sin2(θ)=1

Rearranging this identity, we can isolate cos(θ):

cos(θ)=1sin2(θ)

Since the sine function is bounded between 1 and 1 for all θ, and as θ approaches zero, sin(θ) also approaches zero, we can substitute this limit into our identity:

limθ0cosθ=limθ01sin2(θ)

Given that limθ0sinθ=0, we then have:

limθ01sin2(θ)=102=1

Thus, we confirm that the limit of cos(θ) as θ approaches zero is 1.

the limit of cos(θ)

limθ0cosθ=1

Exploring the Limit of Sine Theta Over Theta

A pivotal limit in calculus, particularly relevant in the study of derivatives and integrals of trigonometric functions, is limθ0sinθθ.
 
To understand this limit, we look to the unit circle, where the sine and tangent functions provide geometric insights into this foundational limit.
The same diagram as the previous one. However, the triangle is expanded. The base is now from the origin to (1,0). The height goes from (1,0) to (1, tan(theta)). The hypotenuse goes from the origin to (1, tan(theta)). As such, the height is now tan(theta). It shows that for 0 < theta < pi/2, sin(theta) < theta < tan(theta).
Figure 8. The sine and tangent functions are shown as lines on the unit circle.

Analyze the behavior of sin(θ) and tan(θ) within the first quadrant of the unit circle, specifically for angles θ where 0<θ<π2.

In this range, it’s clear from the geometric representation that sin(θ) is always less than the length of the tangent line segment from the point on the circle to the x-axis, which is tan(θ). Consequently, we have the inequality:

0<sinθ<tanθ

By dividing each term in the inequality by sinθ , we are led to:

1<θsinθ<1cosθ

With the reciprocal, this inequality can be restated as:

1>sinθθ>cosθ

As θ approaches zero, cos(θ) approaches 1. Therefore, sin(θ) is squeezed between cos(θ) and 1.

Since cos(θ) also approaches 1 as θ  approaches zero, the Squeeze Theorem can be applied to conclude that:

limθ0sinθθ=1

the limit of sinθθ

limθ0sinθθ=1

Evaluating the Limit of 1cosθθ

As we build upon the understanding of limits involving trigonometric functions, the next step is to apply the Squeeze Theorem to evaluate limits that are not immediately obvious. 

In the example below, we use the limit of sinθθ to establish limθ01cosθθ=0. This limit also proves useful in later modules.

Evaluate limθ01cosθθ

the limit of 1cosθθ

limθ01cosθθ=0