Question

In Problems \(41-48\), determine whether the function is continuous at the given point \(c .\) If the function is not continuous, determine whether the discontinuity is removable or non removable. $$ f(x)=\sin x ; c=0 $$

Short answer

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

Step-by-step solution

Understanding Continuity

A function is continuous at a point \(c\) if:\( \lim_{{x \to c}} f(x) = f(c) \). This means the limit of the function as it approaches \(c\) is equal to the function's value at \(c\). We need to check this for \(f(x) = \sin x\) at \(c=0\).

Evaluate \(f(c)\)

First, we calculate the value of the function at \(c=0\). \( f(0) = \sin(0) = 0 \).

Evaluate \(\lim_{{x \to 0}} \sin x\)

Next, find the limit of the function as \(x\) approaches 0. \( \lim_{{x \to 0}} \sin x = \sin(0) = 0 \). The function approaches 0 as \(x\) approaches 0.

Check the Continuity Condition

Compare the limit and the value of the function at \(c\). Since \(\lim_{{x \to 0}} \sin x = f(0) = 0\), the function is continuous at \(c=0\).

Determine the Type of Discontinuity

Since we found \(f(x)\) continuous at \(c=0\), there is no discontinuity. Therefore, there is no removable or non-removable discontinuity to consider here.

Key concepts

Limits

In mathematics, the concept of limits is pivotal for understanding how functions behave as their input approaches a certain value. The idea of a limit is to assess what the value of a function tends to, as the input gets closer to a specific point, without necessarily reaching it. For example, when analyzing the behavior of the function \(f(x) = \sin x\) as \(x\) approaches 0, we use the limit to see if \(f(x)\) smoothly transitions to a certain value, here being 0.

When we say \( \lim_{{x \to c}} f(x) = L \), it means as x gets infinitely close to \(c\) from either side, \(f(x)\) approaches \(L\). It's an essential concept for ensuring continuity. To ensure a function is continuous at a point, the limit must equal the function's value at that point. Understanding limits helps pave the way to mastering more advanced topics in calculus!

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, representing relationships in triangles and circles. These functions include sine, cosine, and tangent among others. Sine, one of these basic functions, is denoted as \( \sin x\). It measures the ratio of the opposite side to the hypotenuse in a right-angled triangle

The sine function oscillates between -1 and 1, creating a smooth, wave-like graph. It's periodic, meaning it repeats its values in regular intervals, specifically every \(2\pi\) radians, or 360 degrees.

• The graph of \( \sin x \) looks like a continuous wave.
• This function is commonly found in problems involving wave behavior, circular motion, and oscillations.

In the given exercise, understanding how \( \sin x \) acts at specific points, like \( x = 0 \), relies heavily on grasping these underlying properties.

Removable Discontinuity

A removable discontinuity at a point in a function's domain occurs when a function is not defined or does not match the limit at that point, but can be "fixed" by redefining the function there. Imagine a graphical function with a gap at a certain value \(c\). If we can fill this gap so the graph becomes seamless, the discontinuity is removable.

For \(f(x) = \sin x\), there is no such issue at \(x = 0\) since the function is already perfectly smooth there. The graph of \(\sin x\) smoothly goes through the origin without any breaks. Therefore, no removable discontinuity exists, and nothing needs adjustment for \(x = 0\).

• This concept often applies to rational functions where a factor cancels out a zero in the denominator.
• Removable discontinuities are common in physics and engineering problems involving limits.

It's essential to be able to identify these situations, as they allow us to redefine functions to better understand their nature at certain points.

Continuous Functions

Continuous functions are those prized in calculus due to their unbroken nature. A function is continuous at a point \(c\) if the following holds:

• The limit of the function as it approaches \(c\) from either direction exists.
• \(f(c)\) is defined.
• \(\lim_{{x \to c}} f(x) = f(c)\)

These ensure the function can seamlessly traverse without jumping or breaking at \(c\). In simpler terms, if you draw it without lifting your pencil, it's continuous at that spot.

In the given problem, \(f(x) = \sin x\) at \(x = 0\) fulfills all the conditions for continuity. The function's value matches its limit, making it a continuous function. Understanding these characteristics is essential for tackling deeper calculus problems, including those involving derivatives and integrals.