17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_y→2 (y²+y-6) / (√(8-y²)-y)

Verified step by step guidance

1. First, substitute y = 2 into the expression to check if the limit results in an indeterminate form. You will find that both the numerator and the denominator evaluate to 0, indicating a 0/0 indeterminate form.
2. Since the limit is in an indeterminate form, l'Hôpital's Rule can be applied. According to l'Hôpital's Rule, if the limit of f(y)/g(y) as y approaches a value results in 0/0 or ∞/∞, then the limit can be evaluated as the limit of f'(y)/g'(y).
3. Differentiate the numerator, f(y) = y² + y - 6, with respect to y. The derivative is f'(y) = 2y + 1.
4. Differentiate the denominator, g(y) = √(8-y²) - y, with respect to y. The derivative is g'(y) = (-y/√(8-y²)) - 1.
5. Now, apply l'Hôpital's Rule by taking the limit of the new fraction formed by the derivatives: lim_y→2 (2y + 1) / ((-y/√(8-y²)) - 1). Substitute y = 2 into this expression to evaluate the limit.

Key Concepts

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 the function's value at points where it may not be explicitly defined. Limits are essential for defining continuity, derivatives, and integrals, forming the backbone of calculus.

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) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately, and then re-evaluating the limit.

Square Root Functions

Square root functions, such as √(8 - y²), are important in calculus as they can introduce complexities in limits and derivatives. Understanding how to manipulate and simplify expressions involving square roots is crucial for evaluating limits, especially when they lead to indeterminate forms or require algebraic manipulation to resolve.