Problem 41 - Calculus Early Transcendentals: Pearson New International Edition

Question

Determine whether the function is continuous at the given point \(c\). If the function is not continuous, determine whether the discontinuity is removable or nonremovable. $$ f(x)=\sin x ; c=0 $$

Step-by-Step Solution

Answer

The function is continuous at \( c = 0 \).

Step 1: Determine Function Definition

The function provided is \( f(x) = \sin x \). This is defined for all real numbers, including \( x = 0 \).

Step 2: Calculate the Limit

To check continuity, we need \( \lim_{x \to 0} f(x) \). Calculate \( \lim_{x \to 0} \sin x \). Since sine is a standard trigonometric function, we know \( \lim_{x \to 0} \sin x = \sin 0 = 0 \).

Step 3: Evaluate the Function at \( c = 0 \)

Evaluate the function directly at \( c = 0 \): \( f(0) = \sin 0 = 0 \).

Step 4: Compare Limit and Function Value

For a function to be continuous at \( c \), the following must be equal: \( \lim_{x \to c} f(x) = f(c) \). Here \( \lim_{x \to 0} f(x) = 0 \) and \( f(0) = 0 \). These are equal.

Step 5: Determine Continuity Type

Since the limit as \( x \) approaches \( 0 \) equals \( f(0) \), the function \( f(x) = \sin x \) is continuous at \( c = 0 \).

Key Concepts

Removable Discontinuity

A removable discontinuity occurs at a specific point in a function where the limit exists, but the function is either not defined or is defined with a different value. This means that if a slight change is made, specifically redefining the value of the function at that point, the discontinuity can be 'removed,' making the function continuous at that spot.
In the context of the exercise with the sine function, we evaluated the function at point \( c = 0 \). The limit, as \( x \) approaches 0, matched the function's value at this point. Hence, the given function \( f(x) = \sin x \) is continuous at \( x = 0 \), and no discontinuity exists. Since there's no discrepancy at \( c \), there's no removable discontinuity here.
Remember, removable discontinuities typically occur in functions that can be adjusted to be continuous by redefining a point or "filling in the hole."

Trigonometric Functions

Trigonometric functions are a key part of mathematics, dealing with the relationships of angles and sides in triangles, as well as periodic phenomena. The sine function, \( \sin x \), is one of the fundamental trigonometric functions and is known for being periodic, with the same repeating values every \( 2\pi \).
These functions are continuous over their entire domain, meaning there are no breaks, jumps, or holes within the values they cover. In other words, trigonometric functions like the sine function are continuous for all real numbers, which means that in evaluations like the one in this exercise, where \( c = 0 \), the function stays consistent and smooth.
Understanding trigonometric functions and their continuity is vital as these concepts are widely applicable in fields such as physics, engineering, and even music and sound theory.

Limit of a Function

The limit of a function at a certain point describes the behavior of the function as the input value approaches that specific point. Calculating limits is crucial in understanding how functions behave, especially at points where they might not be explicitly defined.
For this exercise, evaluating the limit of \( \sin x \) as \( x \to 0 \) gives \( \sin 0 = 0 \). It's important to note that for a function to be continuous at a given point, this limit must equal the function's value at that point.

• If \( \lim_{x \to c} f(x) = f(c) \), the function is continuous at \( c \).
• If the limit exists and matches the function value, the point of evaluation is continuous.

In the case of \( \sin x \), the limit at \( c = 0 \) is the same as its function value, ensuring continuity. Mastery of limits not only helps in assessing function behavior but also aids in solving complex calculus problems efficiently.