Use analytic methods to find the value of lim x→π/4 cos 2x / cos x − sin x.

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Recognize that the problem involves finding the limit of a trigonometric expression as $ x $ approaches $ \frac{\pi}{4} $.

Substitute $ x = \frac{\pi}{4} $ into the expression $ \cos 2x / (\cos x - \sin x) $ to check if it results in an indeterminate form.

Use the double angle identity for cosine: $ \cos 2x = 2\cos^2 x - 1 $ to rewrite the numerator.

Simplify the expression $ \cos x - \sin x $ in the denominator using trigonometric identities, if possible.

Evaluate the limit by substituting $ x = \frac{\pi}{4} $ again after simplification, ensuring the expression is no longer indeterminate.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 context, we are interested in evaluating the limit of the expression as x approaches π/4. Understanding limits is crucial for determining the value of functions at points where they may not be explicitly defined.

Trigonometric Functions

Trigonometric functions, such as cosine and sine, are essential in calculus for analyzing periodic phenomena. In this problem, we are dealing with cos(2x), cos(x), and sin(x), which require knowledge of their properties and values at specific angles, particularly π/4. Familiarity with these functions helps in simplifying and evaluating the limit.

L'Hôpital's Rule

L'Hôpital's Rule is a technique used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. When direct substitution in the limit leads to such forms, this rule allows us to differentiate the numerator and denominator separately. Applying L'Hôpital's Rule may be necessary in this problem if the limit yields an indeterminate form upon evaluation.

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Determine whether the following statements are true and give an explanation or counterexample.

The line x=−1 is a vertical asymptote of the function f(x) =x^2 − 7x + 6 / x^2 − 1.

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Textbook Question

The hyperbolic cosine function, denoted $$\cosh\]\left$(x$\right$)$ , is used to model the shape of a hanging cable (a telephone wire, for example). It is defined as $$\cosh\]\left$(x$\right$)=$\frac{e^{x}$+e^{-x}}{2}$ .