Problem 44 - Calculus Early Transcendentals

Question

Evaluate the following limits. $$\lim _{x \rightarrow \pi / 2} \frac{2 \tan x}{\sec ^{2} x}$$

Step-by-Step Solution

Answer

Question: Evaluate the limit of the given function as x approaches π/2: $$\lim _{x \rightarrow \pi / 2} \frac{2 \tan x}{\sec ^{2} x}$$ Answer: The limit of the function as x approaches π/2 is 0.

Step 1: Rewrite the function using simpler trigonometric functions

We're given the function: $$\frac{2 \tan x}{\sec ^{2} x}$$ Recall that: $$\tan x = \frac{\sin x}{\cos x}$$ $$\sec x = \frac{1}{\cos x}$$ Thus, we can rewrite the given function as: $$\frac{2 \frac{\sin x}{\cos x}}{(\frac{1}{\cos x})^{2}}$$

Step 2: Simplify the expression

Now, let's simplify the function: $$\frac{2 \frac{\sin x}{\cos x}}{\frac{1}{\cos^2x}} = \frac{2 \sin x}{\cos x} \cdot \cos^2x$$ cancel \(\cos x\) from both numerator and denominator: $$= 2\sin x \cos x$$

Step 3: Determine the limit

Now that the function is simplified, we can find the limit as x approaches π/2: $$\lim _{x \rightarrow \pi / 2} 2\sin x \cos x$$ As x approaches π/2, sin x approaches 1 and cos x approaches 0: $$\lim _{x \rightarrow \pi / 2} 2(1)(0)=0$$ So the limit of the function as x approaches π/2 is 0.

Key Concepts

Trigonometric Limits

Trigonometric limits involve evaluating the behavior of trigonometric functions as they approach a certain angle. These are crucial for many calculus problems. Trigonometric functions like sine, cosine, and tangent each have specific properties and rules that dictate their limits. For example, as an angle approaches \( \frac{\pi}{2} \), the behavior of the sine and cosine functions can lead to unique results.
In the case of \( \lim _{x \rightarrow \pi / 2} \frac{2 \tan x}{\sec ^{2} x} \), our goal is to ascertain the value the expression approaches when \( x \to \frac{\pi}{2} \). As identified in the solution, the limit revolves around transforming the given trigonometric functions into more manageable forms, a critical step for solving these limits efficiently. By substituting trigonometric identities, this complex expression is simplified to an easily solvable format. Understanding how trigonometric limits work helps not only in algebraic manipulation but also in establishing how these functions interact near critical points such as 0, \( \pi/2 \), and \( \pi \).

Trigonometric Identities

Trigonometric identities play a pivotal role in simplifying expressions and solving limits. They are equations involving trigonometric functions that are always true for any angle.
Some of the key identities used include:

• Reciprocal identities: For example, \( \sec x = \frac{1}{\cos x} \), which helps transform secant into cosine for easier calculation.
• Quotient identities: Such as \( \tan x = \frac{\sin x}{\cos x} \), enabling us to express tangent in terms of sine and cosine.
• Pythagorean identities: These relate to the squares of sine and cosine, like \( \sin^2 x + \cos^2 x = 1 \).

Applying these identities effectively can drastically reduce the complexity of trigonometric expressions. In our example, these identities helped rewrite \( \frac{2 \tan x}{\sec^2 x} \) into a form suited for limit evaluation. Recognizing and utilizing these relationships is essential for working with limits and trigonometry more generally.

Limit Evaluation Steps

Evaluating limits is a step-by-step process that often involves simplifying expressions and applying specific limit properties. Here are the steps and strategies used to tackle trigonometric limits:

• Simplify the Expression: Start by transforming the function using known identities. This step is crucial to make the overall calculation more manageable. For example, our initial function \( \frac{2 \tan x}{\sec^2 x} \) simplifies to \( 2 \sin x \cos x \).
• Cancel Common Factors: Reduce the expression by canceling common terms. After transforming the function, if the same term appears in the numerator and the denominator, just cancel it out.
• Substitute Limits: Once simplified, substitute the given limit into the expression. For our example, substituting \( \lim _{x \rightarrow \pi / 2} \) results in the evaluation of \( 2 \times 1 \times 0 = 0 \).
• Check for L'Hôpital's Rule: When faced with indeterminate forms like \( \frac{0}{0} \), consider using L'Hôpital's Rule as a tool to find limits more easily.

Following these steps allows for a systematic approach to determining limits, helping address potential obstacles along the way.