Problem 65 - Calculus Early Transcendentals: Pearson New International Edition

Question

Begin by plotting the function in an appropriate window. Your computer may indicate that some of these limits do not exist, but, if so, you should be able to interpret the answer as either \(\infty\) or \(-\infty\). \(\lim _{x \rightarrow 3^{-}} \frac{\sin |x-3|}{x-3}\)

Step-by-Step Solution

Answer

The limit is \(-1\).

Step 1: Understand the Function

We are given the function \( f(x) = \frac{\sin |x-3|}{x-3} \). Our task is to determine the limit of this function as \( x \) approaches 3 from the left, i.e., \( x \to 3^- \).

Step 2: Simplify the Expression for x < 3

Since \( x < 3 \) implies \( |x-3| = 3-x \), we can substitute this into the function to obtain \( f(x) = \frac{\sin (3-x)}{x-3} \). Notice that \( 3-x = -(x-3) \), so this simplifies further to \( f(x) = -\frac{\sin(x-3)}{x-3} \).

Step 3: Apply the Standard Limit Identity

We recognize the expression \( \frac{\sin(x-3)}{x-3} \) which approaches 1 as \( x \rightarrow 3^- \) given that \( x \to 3 \) from the left. The limit identity \( \lim_{u \to 0} \frac{\sin u}{u} = 1 \) can be applied here, where \( u = x-3 \). Thus, \( \lim_{x \to 3^-} -\frac{\sin(x-3)}{x-3} = -1 \).

Step 4: Provide Conclusion

The limit as \( x \to 3^- \) of \( \frac{\sin |x-3|}{x-3} \) is \(-1\). This is because the expression simplifies to a form where we apply the standard limit result with an additional negative sign.

Key Concepts

Limits

Limits are foundational in calculus, providing a way to describe the behavior of a function as it approaches a particular point. In simple terms, a limit examines what value a function approaches as the input approaches a specific number. For instance, consider the limit \(\lim_{x \to 3^-} f(x)\). This notation is asking for the value that \(f(x)\) is approaching as \(x\) gets closer and closer to 3 from the left side. Limits help us understand and predict behaviors in mathematical functions, even when there's not a clear output value at the point itself. Understanding limits is crucial for dealing with more complex calculus topics, such as derivatives and integrals.

Trigonometric Limits

Trigonometric limits involve limits of trigonometric functions like sine, cosine, and tangent. These functions have distinctive properties and periodic behaviors that are important in calculus.One of the most important trigonometric limits to know is \( \lim_{u \to 0} \frac{\sin u}{u} = 1 \). This specific identity is essential in solving limits that involve sine functions. It's utilized when the expression has a form similar to \(\frac{\sin(x-a)}{x-a}\) as \(x\) approaches \(a\). By recognizing this pattern, we apply the identity to find the limit efficiently. In our original problem, understanding trigonometric limits allowed us to quickly solve the limit as \(x\) approached 3 from the left.

Function Analysis

Function analysis involves examining the properties, behavior, and characteristics of functions. It includes looking at how functions behave as inputs change, and often makes use of calculus concepts, such as limits and derivatives, to provide deeper insights.In analyzing the function \(f(x) = \frac{\sin |x-3|}{x-3}\), we see that it combines different mathematical concepts, such as trigonometric functions and absolute value. By breaking down the function and examining each component separately, we can better understand its behavior, especially near critical points.Graphing is also an indispensable tool in function analysis. Seeing a graphical representation helps to visually verify conclusions made through algebraic manipulation.

Absolute Value Function

The absolute value function reflects a number's distance from zero, always giving a non-negative result. It's expressed as \(|x|\), showing its important property that \(|a| = a\) if \(a \geq 0\), else \(-a\).In our original problem, the expression \(|x-3|\) was crucial because it influenced how the function behaved as \(x\) approached 3. Specifically, for \(x < 3\), \(|x-3|\) becomes \(3-x\), since \(x-3\) is negative. This understanding allowed us to simplify and correctly evaluate the limit.Absolute value often introduces piecewise definitions and adjustments depending on input, thus requiring careful attention in calculus problems.