a. Estimate lim x→π/4 cos 2x / cos x − sin x by making a table of values of cos 2x / cos x − sin x for values of x approaching π/4. Round your estimate to four digits.

검증된 단계별 안내

Step 1: Understand the problem. We need to estimate the limit of the function \( \frac{\cos 2x}{\cos x - \sin x} \) as \( x \) approaches \( \frac{\pi}{4} \).

Step 2: Create a table of values for \( x \) approaching \( \frac{\pi}{4} \) from both the left and the right. Choose values such as \( \frac{\pi}{4} - 0.1 \), \( \frac{\pi}{4} - 0.01 \), \( \frac{\pi}{4} + 0.01 \), and \( \frac{\pi}{4} + 0.1 \).

Step 3: For each chosen value of \( x \), calculate \( \cos 2x \) and \( \cos x - \sin x \).

Step 4: Compute the value of the function \( \frac{\cos 2x}{\cos x - \sin x} \) for each \( x \) value in the table.

Step 5: Observe the trend of the function values as \( x \) approaches \( \frac{\pi}{4} \) and estimate the limit by rounding to four decimal places.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. In this case, we are interested in the limit of the function cos(2x) / (cos(x) - sin(x)) as x approaches π/4. Understanding limits is crucial for evaluating the function's value at points where it may not be directly computable.

Trigonometric Functions

Trigonometric functions, such as cosine and sine, are periodic functions that relate angles to ratios of sides in right triangles. In this problem, we are specifically dealing with cos(2x), cos(x), and sin(x). Familiarity with these functions and their properties, including their values at specific angles, is essential for estimating the limit accurately.

Table of Values

Creating a table of values involves calculating the function's output for various inputs approaching a specific point, in this case, π/4. This method provides a visual representation of how the function behaves near the limit and helps in estimating the limit by observing the trend of the values. It is a practical approach to understanding limits when direct substitution may lead to indeterminate forms.