Problem 31 - Single Variable Calculus

Question

Let $$ f(x)=\left\\{\begin{array}{ll} \frac{\sin 2 x}{x} & \text { if } x \neq 0 \\ c & \text { if } x=0 \end{array}\right. $$ Find the value of \(c\) that will make \(f\) continuous on \((-\infty, \infty)\).

Step-by-Step Solution

Answer

The value of \(c\) that makes the function \(f(x)\) continuous on \((-\infty, \infty)\) is \(c = 2\).

Step 1: Identify the Types of Continuity

An essential concept in this problem is three types of continuity: 1. Left-continuous: The function is continuous for all x-values to the left of a point, 2. Right-continuous: The function is continuous for all x-values to the right of a point, 3. Continuous: The function is continuous on its entire domain. In our case, f(x) is given in two parts – one for x ≠ 0 and another for x = 0. So, we will first check if f(x) is left- and right-continuous at x = 0, and then we will analyze the continuity for x ≠ 0.

Step 2: Analyze Continuity at x ≠ 0

Since \(\frac{\sin 2x}{x}\) is a variation of the sinc function, which is continuous for all x ≠ 0, f(x) is continuous for x ≠ 0. Now, our focus should be on the behavior of f(x) around x = 0, specifically the left- and right-continuity.

Step 3: Determine the Left and Right Limits at x = 0

To ensure that f(x) is continuous at x = 0, we need the left limit and right limit as x approaches 0 to be equal. Let's determine both left and right limits. Left limit as x approaches 0: \[\lim_{x \to 0^-} \frac{\sin 2x}{x}\] Right limit as x approaches 0: \[\lim_{x \to 0^+} \frac{\sin 2x}{x}\] Now, notice that both limits are of the same form, so finding one of them would suffice. We will evaluate the right limit using L'Hôpital's Rule since \(\lim_{x \to 0^+} \sin 2x = 0\) and \(\lim_{x \to 0^+} x = 0\): Applying L'Hôpital's Rule: \[\lim_{x \to 0^+} \frac{\sin 2x}{x} = \lim_{x \to 0^+} \frac{\cos 2x \cdot 2}{1} = 2 \cos(0) = 2\] Now that we have determined both left and right limits, the limit of f(x) as x approaches 0 exists and is equal to 2.

Step 4: Determine the Value of c for Continuity

For f(x) to be continuous at x = 0, the following condition must be met: \[\lim_{x \to 0} f(x) = f(0)\] We already know the limit as x approaches 0 is 2: \[\lim_{x \to 0} f(x) = 2\] Since f(0) = c, we have: \[2 = c\] Thus, the value of c that makes f(x) continuous on the given domain is 2.

Key Concepts

Understanding the Sinc Function

The sinc function, denoted as \(\text{sinc}(x)\), plays a pivotal role in understanding the continuity of certain types of functions within calculus. Commonly represented as \(\frac{\sin x}{x}\), the sinc function is defined for all real numbers except for zero, where it is conventionally completed by defining \(\text{sinc}(0)=1\). This is done to ensure the function is continuous across its entire domain.

Applying L'Hôpital's Rule

When faced with an indeterminate form like \(0/0\) or \(\infty/\infty\) in limits, L'Hôpital's Rule provides a powerful tool for finding these limits. The rule states that if \(\lim_{x \to c} f(x)\) and \(\lim_{x \to c} g(x)\) are both zero or both infinity, and the functions \(f\) and \(g\) are differentiable near \(c\) except perhaps at \(c\) itself, then \(\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}\), provided the limit on the right side exists or is infinity.

Calculating Limits for Continuity

In calculus, the concept of limits is crucial to understanding the behavior of functions at specific points or as they approach infinity. Limits describe the value that a function approaches as the input approaches some value. When determining continuity at a point \(x=c\), the function must be:

• Defined at \(c\),
• Approach a specific limit as \(x\) approaches \(c\), and
• Have this limit match the actual value of the function at \(c\).

If all three conditions are met, then the function is continuous at \(x=c\). In the case of the exercise, determining the limit as \(x\) approaches zero and setting it equal to \(f(0)\) allowed for finding the value of \(c\) to ensure the continuity of \(f(x)\) across its domain.