Problem 42 - Thomas Calculus

Question

Find the limits in Exercises \(39-44\) $$ \lim _{x \rightarrow 0} \sin \left(\frac{\pi+\tan x}{\tan x-2 \sec x}\right) $$

Step-by-Step Solution

Answer

-1

Step 1: Identify the Approach

We are asked to find the limit of the function as \(x\) approaches 0. The expression inside the sine function involves trigonometric functions, so first, we need to simplify \(\frac{\pi+\tan x}{\tan x-2 \sec x}\) as \(x\) approaches 0.

Step 2: Expand Trigonometric Functions

Recall the expansions near \(x = 0\): \(\tan x \approx x\) and \(\sec x \approx 1\). Substitute these into the expression: \(\frac{\pi + \tan x}{\tan x - 2 \sec x} \approx \frac{\pi + x}{x - 2}\).

Step 3: Evaluate the Limit of the Simplified Expression

Now, evaluate \(\lim_{x \rightarrow 0} \frac{\pi + x}{x - 2}\). As \(x\) approaches 0, the numerator becomes \(\pi + 0 = \pi\) and the denominator becomes \(0 - 2 = -2\). Therefore, the limit of the expression is \(\frac{\pi}{-2}\).

Step 4: Evaluate the Final Limit

Insert the evaluated limit back into the sine function: \(\lim_{x \rightarrow 0} \sin \left(\frac{\pi}{-2}\right) = \sin(-\frac{\pi}{2})\). We know \(\sin(-\frac{\pi}{2}) = -1\) because \(\sin\) of \(-\frac{\pi}{2}\) corresponds to \(-90\) degrees on the unit circle.

Key Concepts

Trigonometric Limits

Trigonometric limits are essential when dealing with functions that involve trigonometric expressions. These limits often appear in calculus, especially during the examination of behavior near specific points. Take for example the limit we analyzed: \( \lim _{x \rightarrow 0} \sin \left(\frac{\pi+\tan x}{\tan x-2 \sec x}\right) \). Handling such expressions requires a deeper understanding of the behavior of trigonometric functions around specified points. Remember that sinuses, cosines, and tangents have predictable behaviors near zero, allowing us to use approximations to simplify complex limits. This is crucial when the expression would otherwise be complicated or undefined at that point.

Indeterminate Forms

Indeterminate forms can occur when evaluating limits directly gives uncertain results like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). In our original exercise, simplifying \(\lim_{x \rightarrow 0} \frac{\pi + \tan x}{\tan x - 2 \sec x}\) without substitution leads us directly into an indeterminate form. These forms are tricky and cannot be solved straightforwardly as they require additional manipulation or mathematical techniques to resolve. Recognizing an indeterminate form is a key skill, as it signals the need for appropriate methods like L'Hôpital's Rule, algebraic simplification, or series expansion to find the limit.

Simplifying Expressions

Simplifying expressions, especially trigonometric ones, is often necessary to evaluate a limit. Approximating trigonometric functions for small values of \(x\) can be helpful. As seen in the exercise, substituting \(\tan x \approx x\) and \(\sec x \approx 1\) into the expression simplifies it to \(\frac{\pi + x}{x - 2}\). This approximation reduces complexity, allowing us to compute the limit more easily. Remember that for small angles, \(\tan x\) is close to \(x\) and \(\sec x\) near 1, which can transform a previously unsolvable problem into something more manageable. Simplification helps in overcoming indeterminate forms and finding feasible mathematical footing.

Sine Function

The sine function plays a key role during the last steps of our exercise. After simplifying the given expression and finding its limit, we substitute it back into the sine function - in this case, \(\sin\left(\frac{\pi}{-2}\right)\). It's important to have a strong grasp on the sine function’s behavior over the entire unit circle. At an angle of \(-\frac{\pi}{2}\) radians, or \(-90\) degrees, the sine function gives a value of \(-1\). Understanding these key points on the unit circle is fundamental in dealing with trigonometric functions during limit evaluations.

Limit Theorems

Limit theorems form the backbone of evaluating complex expressions. They offer guidelines and properties that facilitate finding the limit of a function. In our context, being familiar with the limit theorems on trigonometric functions helps manage expressions like \(\lim_{x \rightarrow 0}\tan x\) by approximating \(\tan x \approx x\). Limit theorems include the Squeeze Theorem, limits of polynomial functions, and limits involving continuous functions. Each offers a framework to predict and calculate the behavior of functions as they approach certain values seamlessly. They significantly enhance our efficiency and accuracy in calculus problem-solving.