Question

Limits by graphs 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 \(\lim _{x \rightarrow 0} \frac{\tan p x}{\sin x},\) for any real constant \(p\).

Short answer

Based on the given expressions and their limits as x approaches 0, we can observe that the limit of \(\frac{\tan px}{\sin x}\) as x approaches 0 is equal to the constant p. Therefore, our conjecture for a general form is: \(\lim_{x\rightarrow 0} \frac{\tan px}{\sin x} = p\)

Step-by-step solution

Estimate the limits using a graphing utility

To estimate the limits using a graphing utility, input the given expressions into the graphing utility and explore the graph as x approaches 0. For the expressions: 1. \(\frac{\tan 2x}{\sin x}\) 2. \(\frac{\tan 3x}{\sin x}\) 3. \(\frac{\tan 4x}{\sin x}\) Using a graphing utility such as Desmos or Wolfram Alpha, evaluate each expression as x approaches 0: 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 for the general form

Based on the three limits evaluated in Step 1, observe that for each limit, the coefficient of x in the numerator ended up being the resulting limit, while the denominator was always \(\sin x\). Therefore, we can make a conjecture for the general form: \(\lim_{x\rightarrow 0} \frac{\tan px}{\sin x} = p\) In other words, if we take the limit as x approaches 0 of the expression \(\frac{\tan px}{\sin x}\) for any real constant p, the result will be the constant p.

Key concepts

Graphing Utility

In calculus, a graphing utility is a powerful tool to visualize and analyze behavior of mathematical expressions and functions, especially when calculating limits. If you're struggling to picture how a function behaves as it approaches a certain point, a graphing utility like Desmos or Wolfram Alpha can provide a clear visual representation. For example, when you plug in the function \( \frac{\tan 2x}{\sin x} \) into one of these utilities, you can zoom in on the graph near \( x = 0 \) to understand how the function behaves around that point. This visual aid is crucial for comprehending the concept of limits and for estimating their values when the algebra gets tough.
Utilize the interactive features to manipulate parameters and observe the changes in real-time. Besides aiding in understanding, a graphing utility can serve as a check against your algebraic work, reducing errors and building a deeper intuition for the subject matter.

Limit of a Function

The limit of a function is a fundamental concept in calculus that describes the behavior of a function as it approaches a certain point. Basically, when we say \( \lim_{x\rightarrow a} f(x) = L \), we're saying that as \( x \) gets infinitely close to \( a \), the function \( f(x) \) gets infinitely close to a value \( L \). It's important to note that \( x \) doesn't need to equal \( a \)—we're interested in the behavior near \( a \), not necessarily at \( a \).
For example, in our problem with \( \tan(px) \) and \( \sin(x) \), the limits are found by observing the graph as \( x \) approaches zero, not by evaluating the functions at zero. This observation is critical because it leads to understanding that the ratio of \( \tan(px) \) to \( \sin(x) \) simplifies in such a way that only the constant \( p \) remains as the limit.

Tangent and Sine Functions

Let's dive into the specific behaviors of tangent and sine functions. The sine function, \( \sin(x) \), is a periodic wave that oscillates between -1 and 1. Now consider the tangent function, \( \tan(x) = \frac{\sin(x)}{\cos(x)} \), another periodic function that has vertical asymptotes wherever \( \cos(x) = 0 \), due to division by zero. However, as \( x \) approaches 0, \( \tan(x) \) behaves nicely and is simply equal to \( \sin(x) \), if you ignore a scaling factor.
Understanding these behaviors enables you to see why, as \( x \) gets close to 0, the limits of expressions like \( \frac{\tan(px)}{\sin(x)} \) simplify down to just \( p \)—it’s because \( \tan(px) \) and \( \sin(px) \) are behaving similarly at those small values of \( x \) and the complex behavior cancels out, leaving the constant \( p \) as \( x \) approaches 0.

Making Conjectures in Calculus

The process of making conjectures in calculus is all about observing patterns and predicting behavior. When you see a consistent outcome across several examples, as in the limits of \( \frac{\tan(px)}{\sin(x)} \) for different values of \( p \) as \( x \) approaches 0, you can hypothesize a general rule. It's like detective work, where you gather evidence (in this case, from evaluating specific limits) to form a broader conclusion.
A conjecture isn't a proof, but it's an educated guess that can lead to a theorem once verified rigorously. For students, the practice of making conjectures fosters deep analytical thinking and helps in understanding the 'why' and 'how' of mathematics, not just the 'what'. When you form a conjecture, always be ready to test it with more examples or even attempt a formal proof, which is a thrilling adventure in the world of calculus.