Problem 47 - Calculus Early Transcendentals

Question

$$\begin{aligned} &\text {a. Use a graphing utility to estimate } \lim _{x \rightarrow 0} \frac{\tan 2 x}{\sin x}, \lim _{x \rightarrow 0} \frac{\tan 3 x}{\sin x}, \text { and }\\\ &\lim _{x \rightarrow 0} \frac{\tan 4 x}{\sin x} \end{aligned}$$ b. Make a conjecture about the value of \(\lim _{x \rightarrow 0} \frac{\tan p x}{\sin x},\) for any real constant \(p\)

Step-by-Step Solution

Answer

Question: Estimate the limit for the given expression and make a conjecture about the general limit for any real constant p: $$\lim_{x \rightarrow 0} \frac{\tan(px)}{\sin(x)}$$ Answer: The estimated limits for the given expressions are the same as the constant multiples of x inside the tangent function: $$\lim_{x \rightarrow 0} \frac{\tan(px)}{\sin(x)} \approx p$$ for any real constant p.

Step 1: Estimate the limit using a graphing utility

Before estimating the limit for each expression, let's first discuss the properties of the tangent and sine functions near x = 0. The limit will involve the behavior of the function as x approaches 0. As x approaches 0, the sine function can be approximated by its linear term: sin(x) ≈ x The tangent function can be approximated as: tan(x) ≈ x Now we can use a graphing utility to estimate the limit for each expression:

Step 2: Estimate the limit for the first expression

Using a graphing utility, plot the function: $$ f(x) = \frac{\tan(2x)}{\sin(x)} $$ As x approaches 0, the f(x) values approach 2. Therefore, we can estimate the limit as: $$ \lim_{x \rightarrow 0} \frac{\tan(2x)}{\sin(x)} \approx 2 $$

Step 3: Estimate the limit for the second expression

Using a graphing utility, plot the function: $$ f(x) = \frac{\tan(3x)}{\sin(x)} $$ As x approaches 0, the f(x) values approach 3. Therefore, we can estimate the limit as: $$ \lim_{x \rightarrow 0} \frac{\tan(3x)}{\sin(x)} \approx 3 $$

Step 4: Estimate the limit for the third expression

Using a graphing utility, plot the function: $$ f(x) = \frac{\tan(4x)}{\sin(x)} $$ As x approaches 0, the f(x) values approach 4. Therefore, we can estimate the limit as: $$ \lim_{x \rightarrow 0} \frac{\tan(4x)}{\sin(x)} \approx 4 $$

Step 5: Make a conjecture about the general limit

Observe that the limits of each expression are the same as the constant multiples of x inside the tangent function. This suggests that: $$ \lim_{x \rightarrow 0} \frac{\tan(px)}{\sin(x)} \approx p $$ for any real constant p.

Key Concepts

Using Graphing Utility

When working with trigonometric functions, a graphing utility can be an invaluable tool for visualizing behaviors and estimating limits. This is particularly useful when dealing with functions that aren't easily simplified or when a precise analytical solution is challenging to obtain. By plotting the functions \( \frac{\tan(2x)}{\sin(x)} \) and its variations on the graphing utility, students can watch as x approaches 0 and observe the function's value approaching a certain number.

In practice, a graphing utility helps by providing a dynamic visual representation of the functions which can be zoomed in on to observe the behavior around the point of interest, namely x=0 in this exercise. Students feeling overwhelmed by abstract formulas often find solace in the concrete nature of graphical outputs. It's important to note, however, that while graphing utilities are extremely helpful, they are not infallible and should only be used to approximate limits rather than determine exact values.

Tangent and Sine Function Behavior

Understanding the behavior of tangent and sine functions as x approaches zero is critical for approximating limits of trigonometric expressions. The limit process focuses on the function's behavior near a particular point, rather than the value at that point. Both the sine and tangent functions are initially linear as x approaches 0, meaning they can be approximated as \( \sin(x) \approx x \) and \( \tan(x) \approx x \), respectively.

This can be confirmed by looking at their Taylor series expansions where the x term is the first and the most significant when x is very small. Knowing this approximation aids in simplifying the expressions involved in the limit and can transform a complex trigonometric function into an algebraic expression that is much easier to deal with. This characteristic of tangent and sine leads to the estimated limits being the multipliers of x in the tangent, as seen in the textbook exercise.

Approximating Limits

Approximating limits, especially for trigonometric functions, can involve a mix of techniques, including graphing, algebraic manipulation, and understanding of the function's fundamental behavior. In the given exercise, after graphing, we can compare the graphical results with algebraic conjectures. For instance, the proximity of the values of \( \tan(2x) \) and \( \sin(x) \) to 2x and x respectively, as x approaches zero, means that their ratio is expected to approach 2.

This method can be applied to complex functions where a limit cannot be easily found by direct substitution or factoring. Such approximation is only valid close to the point of interest (in this case x=0) and its accuracy diminishes further away from this point. This approach allows students to gain insight into the behavior of more complex functions and understand the foundations of calculus which is built around limits and continuous behavior.

Conjecture in Calculus

Making a conjecture in calculus involves creating a hypothesis based on observed patterns or behaviors. In our textbook exercise, by examining the results of graphing several ratios of \( \tan(px) \) to \( \sin(x) \) as x approaches 0, a pattern emerges. This pattern, a direct correlation between the constant multiplier inside the tangent function and the approximate value of the limit, leads us to conjecture that \( \lim_{x \rightarrow 0} \frac{\tan(px)}{\sin(x)} \approx p \) for any real constant p.

Conjectures are an essential first step in proving theorems. In calculus, conjectures lead to the discovery of relationships and properties which may later be rigorously proven. It's an example of the intuitive aspect of mathematics where observation and logical reasoning come into play. Students should be encouraged to look for patterns and test their conjectures against various functions to deepen their understanding of calculus concepts.