Ch. 2 - Limits and Continuity

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 2 - Limits and Continuity

Problem 2.2.48

Chapter 2, Problem 2.2.48

Limits with trigonometric functions

Find the limits in Exercises 43–50.

limx→0 (1 + x + sin x) / (3 cosx)

Verified step by step guidance

Step 1: Understand the problem. We need to find the limit of the function (1 + x + sin(x)) / (3 cos(x)) as x approaches 0.

Step 2: Apply the limit property that allows us to evaluate the limit of a quotient by finding the limits of the numerator and the denominator separately, provided the limit of the denominator is not zero.

Step 3: Evaluate the limit of the numerator, 1 + x + sin(x), as x approaches 0. Use the fact that sin(x) approaches 0 as x approaches 0, and x approaches 0 as x approaches 0. Therefore, the limit of the numerator is 1 + 0 + 0 = 1.

Step 4: Evaluate the limit of the denominator, 3 cos(x), as x approaches 0. Use the fact that cos(x) approaches 1 as x approaches 0. Therefore, the limit of the denominator is 3 * 1 = 3.

Step 5: Combine the results from Steps 3 and 4 to find the limit of the entire expression. The limit of the quotient is the limit of the numerator divided by the limit of the denominator, which is 1 / 3.

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Key Concepts

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

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are essential for understanding continuity, derivatives, and integrals. In this context, evaluating the limit as x approaches 0 helps determine the behavior of the function near that point.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are periodic functions that relate angles to ratios of sides in right triangles. They play a crucial role in calculus, especially when dealing with limits, derivatives, and integrals involving angles. Understanding their behavior near specific points, like x = 0, is vital for solving limit problems.

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) yields an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator. This rule is particularly useful when dealing with limits involving trigonometric functions.

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Textbook Question

Infinite Limits

Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.

lim x→0 (−1) / (x² (x + 1))

Textbook Question

Finding Deltas Algebraically

Each of Exercises 15–30 gives a function f(x) and numbers L, c, and ε>0. In each case, find the largest open interval about c on which the inequality |f(x)−L| <ε holds. Then give a value for δ>0 such that for all x satisfying 0 < |x−c| < δ, the inequality |f(x)−L| < ε holds.

f(x) = 1/x, L = 1/4, c = 4, ε = 0.05

Textbook Question

Graphing Simple Rational Functions

Graph the rational functions in Exercises 63–68. Include the graphs and equations of the asymptotes and dominant terms.

y = −3/(x − 3)

Textbook Question

Finding Limits of Differences When x → ±∞

Find the limits in Exercises 84–90. (Hint: Try multiplying and dividing by the conjugate.)

lim x → ∞ (√(x² + x) − √(x² − x))

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Textbook Question

Exercises 5–10 refer to the function