Problem 52 - University Calculus: Early Transcendentals

Question

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

Step-by-Step Solution

Answer

The limit is -1.

Step 1: Analyze the Expression

Initially, analyze the expression inside the sine function:\[\frac{\pi + \tan x}{\tan x - 2 \sec x}\]As \( x \to 0 \), both \( \tan x \) and \( \sec x \) approach their small angle approximations, \( \tan x \approx x \) and \( \sec x \approx 1 \). Substitute these approximations into the expression to simplify.

Step 2: Substitute Approximations

Substitute the small angle approximations into the expression:\[\frac{\pi + x}{x - 2 \times 1} = \frac{\pi + x}{x - 2}\]This substitution simplifies the expression within the sine function as \( x \to 0 \).

Step 3: Evaluate the Limit

Now, we evaluate the limit of the simplified expression:\[\lim_{x \to 0} \frac{\pi + x}{x - 2}\]As \( x \to 0 \), the expression becomes \( \frac{\pi + 0}{0 - 2} = \frac{\pi}{-2} = -\frac{\pi}{2} \).

Step 4: Compute Sine of the Limit

Finally, since we want the limit of the sine of this expression as \( x \to 0 \), we compute:\[\sin\left(-\frac{\pi}{2}\right)\]The sine of \(-\frac{\pi}{2}\) is \(-1\). Therefore, the limit is \(-1\).

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)), are fundamental in math, often used to relate angles to ratios of sides in right triangles.
They go beyond geometry and apply to functions of real numbers, where they take cyclical patterns. Beyond triangles, sine and cosine functions exhibit wave-like behaviors, replicating every \( 2\pi \) radians.

• Sine Function: The \( \sin \theta \) gives the y-coordinate of a point on the unit circle.
• Cosine Function: The \( \cos \theta \) provides the x-coordinate of the same point.
• Tangent Function: The \( \tan \theta \) is the ratio \( \frac{\sin \theta}{\cos \theta} \), equivalent to dividing the y-coordinate by the x-coordinate.
• Secant Function: The \( \sec \theta \), defined as \( \frac{1}{\cos \theta} \), gives the reciprocal of the cosine of an angle.

An understanding of these functions is pivotal in calculus, especially when analyzing limits. This exploration extends to their behavior as angles approach zero.

Small Angle Approximations

For very small values of \( x \), certain trigonometric functions can be approximated using simple polynomials.
These approximations reduce complexity when evaluating limits or working with trigonometric expressions.

• \( \sin x \approx x \)
• \( \cos x \approx 1 \)
• \( \tan x \approx x \)
• \( \sec x \approx 1 \)

These approximations are valid due to the Taylor series expansions of these functions at \( x = 0 \). For small \( x \), higher degree terms (like \( x^3 \) for \( \sin x \)) contribute negligible values, making them ignorable in practical computation.
This concept proves particularly useful in simplifying complicated expressions, like the one from the initial exercise, aiding in more straightforward limit evaluation.

Evaluating Limits

Limits help us explore the behavior of functions as inputs approach a specific value.
They are fundamental in calculus and offer insight into continuity, derivatives, and integrals.When evaluating limits involving trigonometric functions, certain techniques can simplify the process:

• Apply direct substitution initially, if feasible.
• Utilize algebraic manipulation to simplify complex expressions.
• Employ trigonometric identities and approximations to assist where direct substitution fails.

In our exercise, by simplifying using small angle approximations, we transformed the complex trigonometric expression into a rational function:\[\frac{\pi + x}{x - 2}\] As \( x \) tended to zero, direct substitution yielded the expression \(-\frac{\pi}{2}\). The sine of this limit value gave the final result of \(-1\).Understanding how to work step-by-step from approximations to direct computations helps build a deeper grasp of limit evaluation in trigonometric contexts.