Ch. 2 - Limits and Continuity

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 2 - Limits and Continuity

Problema 2.5.34

Capitolo 2, Problema 2.5.34

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))

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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 \( x \to 0 \).

Examine the inner function \( \sin x^{1/3} \). As \( x \to 0 \), \( x^{1/3} \to 0 \) and thus \( \sin x^{1/3} \to \sin 0 = 0 \).

Substitute the limit of the inner function into the cosine function: \( \cos(\sin x^{1/3}) \to \cos(0) = 1 \) as \( x \to 0 \).

Substitute the result from the cosine function into the tangent function: \( \tan \left( \frac{\pi}{4} \cdot 1 \right) = \tan \left( \frac{\pi}{4} \right) \).

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.

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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.

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One-Sided Limits

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.

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Intro to Continuity

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.

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Percorso guidato

6:04

Introduction to Trigonometric Functions

Pratica correlata

Domanda del libro di testo

Limits as x → ∞ or x → −∞

The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.

lim x→⁻∞ (³√x − ⁵√x) / (³√x + ⁵√x)

314

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Domanda del libro di testo

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limx→0 (tan 2x) / x

304

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Domanda del libro di testo

At what points are the functions in Exercises 13–30 continuous?

y = cos (x) / x