Problem 66 - 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 \(-\infty\). $$ \lim _{x \rightarrow 3^{-}} \frac{\sin |x-3|}{\tan (x-3)} $$

Step-by-Step Solution

Answer

The limit is \(-1\) as \( x \) approaches 3 from the left.

Step 1: Understanding the Problem

We have the limit \( \lim _{x \rightarrow 3^{-}} \frac{\sin |x-3|}{\tan (x-3)} \). We need to evaluate this limit as \( x \) approaches 3 from the left. The absolute value will consider only non-negative values, and the tangent function may change dramatically around the point where it tends to 0.

Step 2: Simplifying the Expression

Consider when \( x \) approaches 3 from the left, \( |x-3| = 3-x \). Therefore, the expression becomes \( \frac{\sin(3-x)}{\tan(x-3)} \), which further simplifies taking into account that both \( \sin(u) \) and \( \tan(u) \) are standard trigonometric functions near zero.

Step 3: Applying L'Hôpital's Rule

Since we encounter the indeterminate form \( \frac{0}{0} \) at \( x=3 \), apply L'Hôpital's rule: differentiate the numerator and the denominator. This yields:\[ \lim _{x \rightarrow 3^{-}} \frac{-\cos(3-x)}{\sec^2(x-3)} = \lim _{x \rightarrow 3^{-}} -\cos(3-x) \times \cos^2(x-3).\]

Step 4: Evaluating the Limit

As \( x \to 3^- \), \( 3-x \to 0^+ \). Therefore, \( \cos(3-x) \to 1 \) and \( \cos^2(x-3) \to 1 \). Thus, the limit approaches \(-1\times 1^2 = -1\).

Step 5: Plotting the Function

Graph the function \( \frac{\sin |x-3|}{\tan (x-3)} \) to visually confirm the behavior as \( x \to 3^- \). Make sure your window covers near \( x=3 \) to observe the behavior approaching from the left side.

Key Concepts

Trigonometric Limits

Trigonometric limits are essential in calculus involving functions like sine, cosine, and tangent. When working with these functions, especially around specific points like 0, understanding their limiting behavior helps in evaluating complex expressions. For instance, consider the limit \( \lim_{x \to a} \frac{\sin x}{x} = 1 \), a fundamental result frequently used in solving indeterminate forms.
The exercise involves the limit \( \lim _{x \rightarrow 3^{-}} \frac{\sin |x-3|}{\tan (x-3)} \). Since both sine and tangent are periodic with known behavior around 0, they provide a manageable environment for applying L'Hôpital's Rule.

• **Sine**, or \( \sin(\theta) \), oscillates between -1 and 1.
• **Tangent**, \( \tan(\theta) \), grows toward infinity as it nears its asymptote.

Understanding these patterns allows you to predict the behavior and simplify limit problems involving trigonometric expressions efficiently.

Graphing Functions

Graphing functions visually communicates their behavior, making it easier to see how they approach certain values or points. For the limit problem \( \lim _{x \rightarrow 3^{-}} \frac{\sin |x-3|}{\tan (x-3)} \), graphing provides immediate insights into how both the numerator and denominator behave.To create a useful graph, start by plotting key points and observing symmetry or periodicity. Visualizing the tangent's steep rise toward infinity or sine's smooth oscillation helps understand drastic changes without complex calculations.

• Choose an appropriate window that includes points slightly lesser than 3 to observe left-hand behavior.
• Notice how near \( x=3 \), \( \tan(x-3) \) creates sharp changes, which the sine function softens due to its bounded nature.

Graphing complements analytical methods, providing confidence in the numerical conclusions derived.

Indeterminate Forms

Indeterminate forms signal situations where traditional arithmetic fails, often seen as \( \frac{0}{0} \) or similar expressions. In our exercise, as \( x \to 3^- \), \( \frac{\sin(3-x)}{\tan(x-3)} \) initially forms \( \frac{0}{0} \), justifying the application of L'Hôpital's Rule.Indeterminate forms often require special techniques, such as:

• **L'Hôpital's Rule**: Differentiate the top and bottom of the fraction separately then evaluate the limit anew.
• **Algebraic manipulation**: Simplify or transform the expression before re-evaluation.

Identifying and handling these forms correctly ensures accuracy in calculus problems, especially in trigonometric limits where initial forms might appear unsolvable.

Calculating Limits

Calculating limits, especially around points causing indeterminate forms, is a critical skill in calculus. For \( \lim _{x \rightarrow 3^{-}} \frac{\sin |x-3|}{\tan (x-3)} \), meticulous steps ensure accuracy:

1. **Initial Evaluation**: Substitute \( x \) with values near the limit point to understand the form.
2. **Apply L'Hôpital's Rule**: Differentiate numerator and denominator, remembering trigonometric derivatives like \( \frac{d}{dx}\sin(x) = \cos(x) \) and \( \frac{d}{dx}\tan(x) = \sec^2(x) \).
3. **Re-evaluate the Limit**: With derivatives calculated, substitute the x-value again and simplify.
4. **Interpretation**: Recognize if the limit approaches a finite number or \( \pm \infty \).

In this exercise, the limit evaluates to -1, verified by both analytic and visual methods, confirming the reliability of calculations in trigonometric functions approaches.