Find the limits in Exercises 31–40. Are the functions continuous at the point being approached?


lim x → 0 tan (π/4 cos (sin x¹/³))

Verified step by step guidance

1. Identify the function whose limit we need to find: \( \tan \left( \frac{\pi}{4} \cos \left( \sin x^{1/3} \right) \right) \). We are interested in the behavior of this function as \( x \to 0 \).
2. Examine the inner function \( \sin x^{1/3} \). As \( x \to 0 \), \( x^{1/3} \to 0 \) and thus \( \sin x^{1/3} \to \sin 0 = 0 \).
3. Substitute the limit of the inner function into the cosine function: \( \cos(\sin x^{1/3}) \to \cos(0) = 1 \) as \( x \to 0 \).
4. Substitute the result from the cosine function into the tangent function: \( \tan \left( \frac{\pi}{4} \cdot 1 \right) = \tan \left( \frac{\pi}{4} \right) \).
5. Evaluate the tangent function at \( \frac{\pi}{4} \). Since \( \tan \left( \frac{\pi}{4} \right) = 1 \), the limit is 1. The function is continuous at \( x = 0 \) because the limit exists and equals the function value at that point.

Key Concepts

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In this case, we are interested in the limit of the function tan(π/4 cos(sin(x^(1/3)))) as x approaches 0. Understanding how to evaluate limits, especially with trigonometric and composite functions, is crucial for determining the behavior of the function near that point.

Continuity

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. For the given limit, we need to check if the limit as x approaches 0 of tan(π/4 cos(sin(x^(1/3)))) is equal to the function's value at x = 0. Continuity ensures that there are no jumps or breaks in the function at the specified point.

Trigonometric Functions

Trigonometric functions, such as tangent, are periodic functions that relate angles to ratios of sides in right triangles. In this limit, we are dealing with the tangent function, which can exhibit unique behaviors near certain points, particularly at multiples of π/2. Understanding the properties of trigonometric functions, including their limits and continuity, is essential for analyzing the limit in this problem.