Problem 46 - Calculus: Early Transcendentals

Question

(a) By graphing the function \( f(x) = (\tan 4x)/x \) and zooming in toward the point where the graph crosses the \( y \) -axis, estimate the value of \( \displaystyle \lim_{x \to 0}f(x) \). (b) Check your answer in part (a) by evaluating \( f(x) \) for values of \( x \) that approach 0.

Step-by-Step Solution

Answer

The limit is 4.

Step 1: Understanding the Function

We need to find the limit of the function \( f(x) = \frac{\tan 4x}{x} \) as \( x \to 0 \). Notice that this function has a form of \( \frac{0}{0} \), which suggests using a limit approach like L'Hôpital's Rule or evaluating it graphically.

Step 2: Graphing the Function

Graph the function \( f(x) = \frac{\tan 4x}{x} \) with a tool like Desmos or a graphing calculator. Zoom in towards the origin (around \( x = 0 \)) to visually estimate where the graph crosses the y-axis.

Step 3: Estimating Limit from Graph

From the graph, as you zoom in towards \( x = 0 \), observe the behavior of \( f(x) \). The value where the graph seems to cross the y-axis is an estimate of \( \lim_{x \to 0} \frac{\tan 4x}{x} \). It appears that the line is approaching 4.

Step 4: Verify by Substitution

Substitute values of \( x \) that are close to 0 into \( f(x) = \frac{\tan 4x}{x} \). Examples: Evaluate \( f(0.1), f(0.01), \) and \( f(0.001) \) to see if the values approach 4. As \( x \) approaches zero, \( \tan 4x \approx 4x \), thus simplifying the expression to approximately 4.

Step 5: Conclusion

By analyzing the graph and testing with small values of \( x \), we confirm that the estimate is correct. Therefore, \( \lim_{x \to 0} \frac{\tan 4x}{x} = 4 \).

Key Concepts

Graphical Estimation

Graphical estimation is a powerful technique for determining limits, especially when dealing with functions that may appear complicated or where direct calculation is challenging. In the context of calculus limits, graphical estimation involves visualizing the function, such as \( f(x) = \frac{\tan 4x}{x} \), on a graph. By observing the behavior of the function as it approaches a particular point, we can estimate the limit.

To perform graphical estimation, use a graphing tool such as Desmos or a graphing calculator. Plot the function across a range of \( x \) values and focus on the area around the point of interest, in this case, \( x = 0 \). Zooming in on the graph allows you to see where the curve intersects the y-axis. This intersection point gives a visual estimate of the limit.

Keep in mind:

• The closer you zoom in near the point of interest, the more accurate your estimation will be.
• Graphical estimation provides a reasonable approximation, but it should ideally be complemented by analytical methods for confirmation.

L'Hôpital's Rule

L'Hôpital's Rule is an essential tool in calculus for finding limits where direct substitution leads to indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). The rule states that if \( \lim_{x \to c} \frac{f(x)}{g(x)} = \frac{0}{0} \) or \( \frac{\infty}{\infty} \), then \[ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \] given both derivatives exist and \( g'(x) eq 0 \).

In the example of \( f(x) = \frac{\tan 4x}{x} \), directly substituting \( x = 0 \) results in an indeterminate form \( \frac{0}{0} \). By applying L'Hôpital's Rule, we differentiate both the numerator and the denominator:

• Numerator: \( f'(x) = 4 \sec^2(4x) \)
• Denominator: \( g'(x) = 1 \)

Analyzing these derivatives allows us to efficiently calculate the limit. Substituting \( x = 0 \) into the differentiated form gives \( \lim_{x \to 0} \frac{4 \sec^2(4x)}{1} \). We find this limit equals 4, confirming the value obtained through graphical estimation.

Keep in mind that L'Hôpital's Rule can simplify complex limit problems by transforming them into more manageable forms. However, ensure that conditions for the rule are satisfied before applying it.

Trigonometric Limits

Trigonometric limits are a specific subset of limit problems that involve trigonometric functions. Understanding these can greatly simplify the process of solving limits involving sine, cosine, tangent, and other trigonometric functions.

For the function \( f(x) = \frac{\tan 4x}{x} \), we encounter a classic trigonometric limit. These types of problems often exploit limit identities like:

• \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \)
• \( \lim_{x \to 0} \frac{\tan x}{x} = 1 \)

Knowing that \( \tan x \approx x \) as \( x \to 0 \), we can substitute \( \tan 4x \approx 4x \) reducing our function to \( f(x) \approx 4 \). Thus, \( \lim_{x \to 0} \frac{\tan 4x}{x} = 4\).

These trigonometric identities are extremely useful when algebraic manipulation alone doesn't simplify things easily. Understanding them and knowing when and how to apply them can make determining limits straightforward. Always remember to verify these with other methods to ensure accuracy of your results.