Problem 52 - AP Calculus with 8 Practice Tests 15th

Question

\(\lim _{x \rightarrow 0} \frac{\sec x-\cos x}{x^{2}}\) \((\mathrm{A})=0\) \((B)=\frac{1}{2}\) \((C)=1\) \((\mathrm{D})=2\)

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Understand the Problem

We need to find the limit of the function \( \frac{\sec x - \cos x}{x^2} \) as \( x \) approaches 0.

Step 2: Recall Trigonometric Identities

Remember that \( \sec x = \frac{1}{\cos x} \). We will use this identity to rewrite the expression.

Step 3: Substitute the Identity

Substitute \( \sec x \) with \( \frac{1}{\cos x} \) to rewrite the limit as follows: \[ \lim_{x \to 0} \frac{\frac{1}{\cos x} - \cos x}{x^2} \]

Step 4: Simplify the Expression

Combine the terms in the numerator over a common denominator: \[ = \lim_{x \to 0} \frac{1 - \cos^2 x}{x^2 \cos x} \] Recall that \( 1 - \cos^2 x = \sin^2 x \), so our limit becomes: \[ \lim_{x \to 0} \frac{\sin^2 x}{x^2 \cos x} \]

Step 5: Evaluate the Limit Using Known Limits

We know that \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \). Therefore, we can replace \( \sin^2 x \) with \( (\sin x)^2 \) to rewrite the limit as: \[ \lim_{x \to 0} \frac{(\sin x)^2}{x^2 \cos x} \] This is equivalent to: \[ \left( \lim_{x \to 0} \frac{\sin x}{x} \right)^2 \cdot \lim_{x \to 0} \frac{1}{\cos x} \]

Step 6: Use Known Limits for Calculation

Calculate each part: - The first part is \( \left( \lim_{x \to 0} \frac{\sin x}{x} \right)^2 = 1^2 = 1 \).- The second part is \( \lim_{x \to 0} \frac{1}{\cos x} = \frac{1}{\cos(0)} = 1 \).Multiply the results: \[ 1 \cdot 1 = 1 \]

Step 7: Conclude the Limit Value

The limit value is the product from Step 6, which is 1. Therefore, the solution to the limit \( \lim_{x \to 0} \frac{\sec x - \cos x}{x^2} \) is \( 1 \).

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in calculus, helping us manipulate and simplify expressions in limits, derivatives, and integrals. A key identity used in the given problem is the definition of the secant function:

• \( \sec x = \frac{1}{\cos x} \).

This identity allows us to transform expressions involving secant into ones that involve cosine, which are often easier to handle.
In the original exercise, substituting \( \sec x \) with \( \frac{1}{\cos x} \) was vital in simplifying the expression \( \frac{\sec x - \cos x}{x^2} \). This strategy helps in recognizing possible simplifications and applying limits, given that expressions involving cosine and sine have well-known limits as \( x \) approaches 0.
Understanding trigonometric identities provides a significant foundation for solving many calculus problems efficiently, making it easier to take limits, differentiate, or integrate trigonometric functions.

Simplification Techniques

When working with limits, simplifying expressions is often necessary to evaluate them correctly. A key example from the solution is simplifying the numerator \( \sec x - \cos x \) to an expression in terms of sine and cosine functions.

• First, substitute \( \sec x \) with \( \frac{1}{\cos x} \) to obtain a common denominator.
• Simplify to \( \frac{1 - \cos^2 x}{x^2 \cos x} \).
• Recognize that \( 1 - \cos^2 x = \sin^2 x \), giving us \( \frac{\sin^2 x}{x^2 \cos x} \).

This transformation uses the Pythagorean identity:

• \( \sin^2 x + \cos^2 x = 1 \).

Such simplifications are crucial as they transform complex trigonometric expressions into forms that make limit evaluation feasible.
In general, the practice of identifying useful identities and rewriting expressions accordingly is a critical skill that enhances your ability to compute limits and solve related calculus problems.

Evaluating Limits

Evaluating limits is a key aspect in understanding the behavior of functions as they approach specific points. Calculating the limit as \( x \to 0 \) for the function \( \frac{\sin^2 x}{x^2 \cos x} \) involves breaking down the expression into simpler parts.
Use these steps:

• Recognize known limits such as \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \).
• Rewrite \( \sin^2 x \) as \( (\sin x)^2 \), allowing us to express the limit as a product of separate limits.
• Evaluate the resulting expression: \( \left( \lim_{x \to 0} \frac{\sin x}{x} \right)^2 \times \lim_{x \to 0} \frac{1}{\cos x} \).
• The known limit \( \lim_{x \to 0} \frac{1}{\cos x} = \frac{1}{\cos(0)} = 1 \).

Combining these calculations yields the final result:

• \( 1 \times 1 = 1 \), indicating the original limit is 1.

Understanding how to decompose and evaluate complex limit expressions using known results is an invaluable skill in calculus.