Question

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

Short answer

Answer: Based on the results from the given limits, we conjecture that the value of the limit \(\lim _{x \rightarrow 0} \frac{\tan nx}{\sin x}\) for any real constant 'n' will always be equal to 'n'.

Step-by-step solution

Estimate the Given Limits Using a Graphing Utility

By using the graphing utility (like Desmos, Geogebra, or a graphing calculator), we need to find the value of each limit as x approaches 0: 1. \(\lim _{x \rightarrow 0} \frac{\tan 2x}{\sin x}\) 2. \(\lim _{x \rightarrow 0} \frac{\tan 3x}{\sin x}\) 3. \(\lim _{x \rightarrow 0} \frac{\tan 4x}{\sin x}\) By observing the graphs and calculating the limits using the graphing utility, we find that: 1. \(\lim _{x \rightarrow 0} \frac{\tan 2x}{\sin x} = 2\) 2. \(\lim _{x \rightarrow 0} \frac{\tan 3x}{\sin x} = 3\) 3. \(\lim _{x \rightarrow 0} \frac{\tan 4x}{\sin x} = 4\)

Make a Conjecture About the General Case

Based on the results we calculated in Step 1, we can notice that each limit is equal to the constant multiplying x inside the tangent function. So, we can make a conjecture for the general case as: For any real constant \(n:\) \(\lim _{x \rightarrow 0} \frac{\tan nx}{\sin x} = n\) Hence, we conjecture that the limit will always be equal to the real constant n, which is the coefficient of x inside the tangent function, as x approaches 0.

Key concepts

Conjecture

In mathematics, a conjecture is an educated guess or a hypothesis based on previous observations or known data. When we want to predict the behavior of certain functions in calculus, forming a conjecture becomes a useful tool. In this exercise, we estimate the limits of expressions involving the tangent and sine functions at points near zero. By calculating specific cases like \( \lim _{x \rightarrow 0} \frac{\tan 2x}{\sin x} = 2 \) and similar forms, patterns emerge.

From these observations, we make a conjecture for a general expression \( \lim _{x \rightarrow 0} \frac{\tan nx}{\sin x} \), where \(n\) is any real number. The prediction is that as \(x\) approaches zero, this limit equals \(n\). Conjectures are proven through rigorous proofs in advanced mathematics, but they start as clever insights from observed patterns.

Graphing Utility

A graphing utility is a software or device that assists in visualizing mathematical functions and their behaviors, making them invaluable in calculus. Tools like Desmos or Geogebra, as well as traditional graphing calculators, allow students to estimate limits and understand function behavior visually.

For complex functions, a graphing utility can plot the curve representing the mathematical expression. By zooming in around the point of interest, in this case, \(x = 0\), we can observe how the function behaves. As we analyzed \( \frac{\tan nx}{\sin x} \) near zero using a graphing utility, it suggested that these expressions converge to values equaling \(n\). This technological aid helps verify calculations and supports forming reliable conjectures by offering visual and numerical insights.

Trigonometric Functions

Trigonometric functions, like sine and tangent, are fundamental in calculus and represent relationships between angles and lengths in right-angled triangles. These periodic functions have unique features, such as oscillation patterns, which are crucial in various mathematical and physical applications.

In this context, we examine the behavior of \( \tan nx \) and \( \sin x \) as \( x \to 0 \). The sine function starts linear near zero, while the tangent function, expressed here through \( \tan(nx) \), starts with steeper slopes influenced by \(n\). Understanding these functions' derivatives and series expansions helps us comprehend the outcome of their limit behavior, showing that these primary trigonometric functions' ratios lead to the conjecture \(n\). Recognizing these properties deepens comprehension of how trigonometric functions connect with calculus limits.

Tangent and Sine Relations

The tangent and sine functions are closely related through their geometric definitions and unit circle properties. In calculus, analyzing these function combinations, like in the limit \( \frac{\tan nx}{\sin x} \), reveals interesting insights.

Near \(x = 0\), \(\sin x\) approximates to \(x\) and \(\tan x\) approximates to \(x\) as well, making the ratio \(\frac{\tan x}{\sin x}\) tend towards 1. When scaled by a factor of \(n\) inside the tangent, \(\tan(nx)\) scales the behavior linearly by \(n\), emphasizing that the limiting process primarily depends on this coefficient.

• Understanding these approximations helps demystify how functions blend at specific points.
• Working with these calculations offers a clearer picture of how calculus models allow us to make generalized predictions about function behaviors.

The relationship between tangent and sine underscores how algebraic manipulations in calculus reveal deeper properties of trigonometric functions.