Ch. 3 - Derivatives

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

Hass 15th Edition

Ch. 3 - Derivatives

Problem 3.5.51

Chapter 3, Problem 3.5.51

Is there a value of c that will make

f(x) = { (sin²(3x)) / x², x ≠ 0
c, x = 0

continuous at x = 0? Give reasons for your answer.

Verified step by step guidance

To determine if the function f(x) is continuous at x = 0, we need to check if the limit of f(x) as x approaches 0 is equal to f(0). This means we need to find the limit of (sin²(3x)) / x² as x approaches 0.

Apply the limit definition: lim(x→0) (sin²(3x)) / x². Notice that this is an indeterminate form 0/0, so we can use L'Hôpital's Rule, which states that if the limit of f(x)/g(x) as x approaches a point is indeterminate, then it is equal to the limit of f'(x)/g'(x) as x approaches the same point, provided the derivatives exist.

Differentiate the numerator and the denominator separately. The derivative of sin²(3x) with respect to x is 2sin(3x)cos(3x) * 3, using the chain rule. The derivative of x² with respect to x is 2x.

Apply L'Hôpital's Rule: lim(x→0) [2sin(3x)cos(3x) * 3] / [2x]. Simplify the expression to find the limit as x approaches 0.

Once the limit is found, set it equal to c, since f(x) must be equal to f(0) for continuity at x = 0. This will give you the value of c that makes the function continuous at x = 0.

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Key Concepts

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

Continuity at a Point

A function is continuous at a point if the limit of the function as it approaches the point from both sides equals the function's value at that point. For f(x) to be continuous at x = 0, the limit of f(x) as x approaches 0 must equal f(0), which is c in this case.

Limit of a Function

The limit of a function describes the behavior of the function as the input approaches a particular value. To find the limit of f(x) = (sin²(3x)) / x² as x approaches 0, we need to evaluate the expression's behavior near x = 0, often using techniques like L'Hôpital's Rule or trigonometric identities.

L'Hôpital's Rule

L'Hôpital's Rule is a method for finding limits of indeterminate forms like 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) as x approaches a value results in an indeterminate form, the limit can be found by differentiating the numerator and denominator separately and then taking the limit again.

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