Question

Think About It When using a graphing utility to generate a table to approximate $$ \lim _{x \rightarrow 0} \frac{\sin x}{x} $$ a student concluded that the limit was 0.01745 rather than 1 Determine the probable cause of the error.

Short answer

The mistake likely happened because the student used degrees instead of radians in their graphing utility for the trigonometric functions. The limit \(\lim _{x \rightarrow 0} \frac{\sin x}{x}\) is actually equal to 1.

Step-by-step solution

Identify the correct limit

The limit \(\lim _{x \rightarrow 0} \frac{\sin x}{x}\) is a well-known standard limit in Calculus, which equals 1.

Identify the student's mistake

The student likely made a common error when using their graphing utility by assuming the trigonometric functions were set to operate in degrees rather than radians, which is what many calculators and softwares default to.

Calculate \(\frac{\sin x}{x}\) with x in degrees

To verify this hypothesis, we can calculate \(\frac{\sin x}{x}\) at a small value of \(x\), but this time, take \(x\) to be in degrees. This is equivalent to finding \(\frac{\sin x_{rad}}{x_{deg}}\), where \(x_{rad} = x_{deg}\pi/180\). If we let \(x_{deg} = 1\degree\), then \(x_{rad} = \pi/180\approx 0.01745\), which is the value the student got.

Key concepts

Limits

Limits are fundamental in calculus, allowing us to explore the behaviour of functions as inputs approach certain values. A limit describes the value that a function approaches as the input, or the variable in it, "approaches" some value. In formal terms, \[\lim_{x \to a} f(x) = L \]means that as \(x\) gets closer and closer to \(a\), \(f(x)\) gets closer to \(L\). In this exercise, the limit we are interested in is \[\lim_{x \rightarrow 0} \frac{\sin x}{x}\],which equals 1. This specific limit is a classic example often used to illustrate this concept, particularly because \(\sin x\) and \(x\) approach zero simultaneously, causing standard division to be indeterminate (\(0/0\)). Evaluating this requires careful consideration of how \(\sin x\) behaves around zero, revealing the need for understanding trigonometric behaviour near this point, an aspect expanded in calculus.

Trigonometric functions

Trigonometric functions, such as sine, cosine, and tangent, are crucial in mathematics for their ability to model periodic phenomena. These functions relate the angles of a triangle to the lengths of its sides. The sine function, \(\sin x\), is especially relevant in calculus when examining limits as it includes important properties for such situations.

• The sine function can be understood as the y-coordinate of a point on the unit circle.
• It oscillates between -1 and 1, and is periodic with period \(2\pi\). This makes it valuable for various applications, from physics to engineering.
• Understanding its behaviour near zero is key to solving the limit problem as the value approaches it.

Grasping the subtleties of these functions is essential for correctly interpreting their values, especially when expected computational outputs deviate from actual values due to missteps, like mixing radian and degree measurements.

Radians vs Degrees

Radians and degrees are units of measure for angles, and understanding their application is critical in calculus and other fields that require angle measurement.While degrees are the more familiar unit in general education and everyday usage, calculus conventionally uses radians because they simplify mathematical problems, particularly derivatives and integrals of trigonometric functions.

• A degree is 1/360th of a full circle.
• A radian, on the other hand, is the angle subtended at the center of a circle by an arc whose length is equal to the circle's radius.

One full circle is \(360\degree = 2\pi\) radians. This relationship means:

• \(\pi\) radians = \(180\degree\)
• \(1\degree = \frac{\pi}{180}\) radians
• \(1\) radian is about \(57.2958\degree\)

The error in the student's calculation likely stems from not switching to radians from degrees, which causes computational discrepancies.

Graphing utility

Graphing utilities are tools like calculators or software applications used to visualize functions and computations in mathematics. These tools enhance the understanding of complex functions by providing visual representations and numerical approximations. However, proper usage is critical.When using a graphing utility:

• Always verify the angle measurement settings; ensure you are operating in radians unless otherwise required. This was crucial in the student's error finding \(\sin x/x = 0.01745\) instead of 1 when incorrectly using degrees.
• These utilities provide an excellent approximation for limits if correctly configured.
• They can plot functions over a range, helping users to observe behaviour and confirm analytical work.

Despite their effective capabilities, reliance on graphing utilities requires checking settings such as angle measurement units, as errors here can lead to significant misunderstanding in calculations.