Problem 39 - Precalculus

Question

For the following exercises, use a calculator to estimate the limit by preparing a table of values. If there is no limit, describe the behavior of the function as \(x\) approaches the given value. $$\lim _{x \rightarrow 0} \frac{7 \tan x}{3 x}$$

Step-by-Step Solution

Answer

The limit is \( \frac{7}{3} \).

Step 1: Understand the Function

We are given the function \( f(x) = \frac{7 \tan x}{3x} \). We need to find the limit of \( f(x) \) as \( x \) approaches 0. The function involves the tangent function, \( \tan x \), which can be approximated using its Taylor series expansion near 0.

Step 2: Substitute with Taylor Series

For small values of \( x \), \( \tan x \approx x + \frac{x^3}{3} + \cdots \). Thus, the function \( f(x) \) can be approximated as \( \frac{7(x + \frac{x^3}{3})}{3x} \). Simplifying this gives \( \frac{7}{3} + \frac{7x^2}{9} \).

Step 3: Simplify and Analyze the Function

As \( x \to 0 \), the term \( \frac{7x^2}{9} \to 0 \). Therefore, \( f(x) = \frac{7}{3} + \frac{7x^2}{9} \) simplifies to \( \frac{7}{3} \) when considering the limit. Thus, the behavior as \( x \to 0 \) is that \( f(x) \to \frac{7}{3} \).

Step 4: Verify with a Calculator

Use a calculator to evaluate \( \frac{7 \tan x}{3x} \) for values of \( x \) close to 0: try \( x = 0.1, 0.01, 0.001, -0.1, -0.01, -0.001 \). Observing these values, you'll notice that as \( x \) gets closer to 0, \( \frac{7 \tan x}{3x} \) approaches approximately \( 2.3333 \), which is \( \frac{7}{3} \).

Key Concepts

Tangent Function

The tangent function, often written as \( \tan x \), is a fundamental trigonometric function. It relates the angle \( x \) of a right triangle to the ratio of the opposite side over the adjacent side.
- For small angles near zero, the tangent function behaves nearly linearly; \( \tan x \approx x \).- As \( x \) approaches \( \pi/2 \) or \(-\pi/2 \), \( \tan x \) grows significantly, heading towards infinity or negative infinity. However, near zero, where our problem lies, its behavior is quite tame.
- The tangent function can be approximated using a Taylor series, which helps to estimate its value without requiring exact computation. This approximation allows for simplifying complex functions like \( \frac{7 \tan x}{3x} \), by expanding \( \tan x \) near zero and simplifying its components.

Taylor Series

When dealing with trigonometric functions like \( \tan x \) at values near zero, Taylor Series come in handy. A Taylor series expresses a function as an infinite sum of terms calculated from the values of its derivatives at a single point.
The Taylor series expansion for \( \tan x \) around \( x = 0 \) is given by:

• \( \tan x = x + \frac{x^3}{3} + \frac{2x^5}{15} + \cdots \)

For approximation, especially in calculating limits, we often only need a few terms of the series. In our exercise, we substituted \( \tan x \) with its first two terms: \( x + \frac{x^3}{3} \).
- This makes it easier to understand and calculate the behavior of \( \frac{7 \tan x}{3x} \) as \( x \to 0 \).- Substituting in the Taylor series helps us cancel terms and find that the limit of the entire function simplifies to a straightforward value.

Calculator Usage

Estimating limits can often be made more intuitive using a calculator. In this exercise, calculating values of the function \( \frac{7 \tan x}{3x} \) for points very close to zero can help confirm our theoretical estimation. Here's how you can approach this:

• Select values of \( x \) that are close to zero, such as 0.1, 0.01, and 0.001.
• Input these values into your calculator to compute \( \frac{7 \tan x}{3x} \).


• Observe the output; as \( x \) gets smaller, the output should converge towards \( \frac{7}{3} \).

- This hands-on approach using a calculator gives you practical confirmation of your answer.- It also demonstrates the behavior of trigonometric functions empirically, reinforcing your understanding of limits and approximations.
Remember, while calculators are helpful for verification, understanding the underlying mathematics is essential for deeper learning.