Problem 82 - Precalculus: Functions and Graphs

Question

As \(x \rightarrow 0^{+}, f(x) \rightarrow L\) for some real number \(L\) Use a graph to predict \(L\) $$f(x)=\frac{x+\tan x}{\sin x}$$

Step-by-Step Solution

Answer

The predicted limit \( L \) is 2.

Step 1: Understand the Function

We need to analyze the function \( f(x) = \frac{x + \tan x}{\sin x} \) as \( x \) approaches \( 0^+ \). This requires understanding the behavior of \( x + \tan x \) and \( \sin x \) independently as \( x \) approaches \( 0 \).

Step 2: Identify Known Limits

As \( x \rightarrow 0 \), \( \tan x \approx x \) and \( \sin x \approx x \). Thus, we know that \( \tan x \) and \( \sin x \) each approaches 0 with \( x \) and can approximate \( f(x) \).

Step 3: Use Graphing Technology

Utilize graphing technology to plot \( f(x) = \frac{x + \tan x}{\sin x} \) near \( x = 0 \) in order to observe its behavior and make predictions. Note smoothness and any visible limit at \( x \rightarrow 0^+ \).

Step 4: Simplify and Approximate the Limit

Use the approximation \( \tan x \approx x \) as \( x \rightarrow 0 \). So, \( f(x) = \frac{x + \tan x}{\sin x} \approx \frac{x + x}{x}\), which simplifies to approximately 2.

Step 5: Validate Predictions with Calculus

Using L'Hôpital's Rule could validate the limit prediction: differentiate numerator \( f'(x) = 1 + \sec^2(x) \) and denominator \( f'(x) = \cos(x) \). At \( x \approx 0, \sec^2(x) \approx 1 \), confirming \( L' = 2 \).

Key Concepts

Trigonometric Limits

Trigonometric limits are essential in understanding the behavior of trigonometric functions as they approach certain values. In calculus, particularly when evaluating the limit of a function like \( f(x) = \frac{x + \tan x}{\sin x} \), it's crucial to recognize behavior patterns of basic trigonometric functions such as \( \tan x \approx x \) and \( \sin x \approx x \) as \( x \to 0 \). These approximations simplify calculations and help predict how these functions interact near critical points.

Trigonometric limits often involve shortcuts and approximations:

• Small Angle Approximations: For small values of \( x \), \( \sin x \approx x \), \( \cos x \approx 1 \), and \( \tan x \approx x \) can be assumed. These approximations can simplify complex functions.
• Limit Theorems: These include standard limits such as \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \), and \( \lim_{x \to 0} \frac{\tan x}{x} = 1 \), which are foundational in trigonometric limit concepts.

By utilizing these basic approximations, you can simplify and deduce limits efficiently, providing foundational understanding for more advanced calculus topics.

L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool used in calculus for finding the limits of indeterminate forms such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). When faced with the limit \( \lim_{x \to 0} \frac{x + \tan x}{\sin x} \), the function initially presents as an indeterminate form, making this rule essential.

How does L'Hôpital's Rule work? It starts by differentiating the numerator and the denominator separately:

• Differentiate \( x + \tan x \) to get \( 1 + \sec^2(x) \).
• Differentiate \( \sin x \) to obtain \( \cos x \).

You then re-evaluate the limit using these derivatives instead of the original function. Applying this to our function, the limit becomes:

\[\lim_{x \to 0^+} \frac{1 + \sec^2(x)}{\cos x}\]
Expanded further, since \( \sec^2(x) \to 1 \) as \( x \to 0 \), the expression simplifies to 2. Thus, L'Hôpital's Rule confirms our earlier prediction, proving it a vital tool for solving limits in calculus.

Graphical Analysis of Functions

Graphing functions is a practical approach to understanding limits and function behavior visually. By plotting the function \( f(x) = \frac{x + \tan x}{\sin x} \) as \( x \to 0^+ \), you can observe its behavior and make informed predictions about the function's limit.

When analyzing graphs:

• Notice the smoothness of the curve near the critical point. A continuous and smooth curve indicates that the function approaches a particular value or limit smoothly.
• Identify any asymptotic behavior. In cases where the function doesn't touch an axis or line, this might be a key to understanding complex limits.
• Look for horizontal lines that imply horizontal asymptotes, signaling where the function levels out at a particular limit.

Using graphing technology, you can visually confirm the limit value by seeing that the function approaches \( L = 2 \) in this exercise. Combining this approach with algebra and calculus ensures a thorough understanding of function limits.