Ch. 2 - Limits and Continuity

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

Hass 15th Edition

Ch. 2 - Limits and Continuity

Problem 2.5.40

Chapter 2, Problem 2.5.40

Find the limits in Exercises 31–40. Are the functions continuous at the point being approached?
lim ϴ → 0 cos (πϴ/sin ϴ)

Verified step by step guidance

First, recognize that the limit involves a trigonometric function as ϴ approaches 0. The expression is cos(πϴ/sinϴ).

To evaluate the limit, consider the behavior of the argument of the cosine function, πϴ/sinϴ, as ϴ approaches 0.

Use the trigonometric identity sinϴ ≈ ϴ when ϴ is near 0. This approximation helps simplify the expression πϴ/sinϴ to π, since sinϴ/ϴ approaches 1.

Substitute the simplified argument back into the cosine function: cos(πϴ/sinϴ) becomes cos(π) as ϴ approaches 0.

Evaluate cos(π) to determine the limit. Then, consider the definition of continuity: a function is continuous at a point if the limit as ϴ approaches that point equals the function's value at that point. Check if this condition holds for the given function at ϴ = 0.

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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. It helps in understanding how functions behave near points of interest, including points of discontinuity. In this case, we are interested in the limit of the function as θ approaches 0, which requires evaluating the function's behavior close to 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 determine if the function cos(πθ/sin(θ)) is continuous at θ = 0. This involves checking if the limit exists and matches the function's value at that point.

Trigonometric Limits

Trigonometric limits often involve special techniques or identities to evaluate limits involving sine and cosine functions. In this case, the limit involves the sine function in the denominator, which approaches 0 as θ approaches 0. Understanding how to manipulate trigonometric expressions and apply L'Hôpital's Rule or Taylor series can be crucial for finding the limit accurately.

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