17–83. Limits Evaluate the following limits. Use l’Hôpital’s R...

Question

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)

Accepted Answer

Welcome back everyone. Lilap those rules to find the limit of the following function as X approaches -3. We're given a function F of X and for answer choices. A says -25s, B-52s, C 25, and D 5 halves. So what we have to do is first we'll write down the limit limit as X approaches -3 of F of X. Now F of X is X2 + x minus 6 divided by square root of 18 minus X2 + x. We always begin with direct substitution, so we're going to get -32+. -3 minus 6 divided by square root of 18 minus -3 squad. Plus -3 if we evaluate each. Part of the fraction we're going to get 0 divided by 0. And for this form or infinity divided by infinity, we can apply Lapto's rule. Lapto's rule says that we have to differentiate the numerator and the denominator. Let's begin with the numerator. We can apply the power rule, which gives us. The derivative of X2, right? So the derivative of that term is 2 X and the derivative of X is 1. The derivative of -6 is 0. Now let's identify the derivative of the radical term. We have to recall that the derivative of square root of X is 1 divided by 2 square root of X. So we're going to get 1 divided by 2 square root of. In this case, we have 18 minus X squared, and now because this is not just an independent variable, but in inner function, we also have to apply the chain rule and multiply by the derivative of 18 minus. X squad, right? So the derivative of that is 0 minus 2 X, so negative 2 X, right? And now the derivative of X is 1, we're adding 1. Now, let's simplify. We're going to get limit as X approaches -3 of 2 X plus 1. Divided by 1 minus x divided by square root of 18 minus X2 and now once again we can apply direct substitution which gives us 2 multiplied by -3 + 1. Divided by 1 minus -3, so we can say 1 + 3. Divided by square root of 18 minus -32. Now let's evaluate the numerator and the denominator. For the numerator, we're going to get -6 + 1. So that's -5. And now for the denominator we have 1 plus we divided by square root of 9, right? So that's 1 + 1 which gives us 2. So now this is a finite number, meaning we have arrived at the final answer which corresponds to the answer choice B-5 halves. Thank you for watching.

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)

Key Concepts

Limits

L'Hôpital's Rule

Square Root Functions

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