Use l’Hôpital’s rule to find the limits in Exercises 7–52.
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Question

Use l’Hôpital’s rule to find the limits in Exercises 7–52.

37. lim (y → 0) (√(5y + 25) - 5) / y

Accepted Answer

Welcome back, everyone. Compute the following limit using Lapital's rule. Limit as T approaches 0 of square root of 8 T + 16 minus 4 divided by C. So for this problem, let's go ahead and rewrite the limit. Limit is C approaches 0. Of square root of 8 T + 16 minus 4 divided by T. Let's remember that the first step is to assume that our function is continuous at T equals 0. So, we're going to substitute C equals 0 to get square root of 16 minus 4 in the numerator and 0 in the denominator. So, overall, we can simplify and show that this is 0 divided by 0, which is an indeterminate form, and it is one of the indeterminate forms that is suitable for Lapto's rules to be applied. Meaning we can go ahead and apply Lappito's rule. It tells us that we want to find the derivative of the numerator. And the derivative of the denominator. So, let's begin with the numerator. In the numerator, we have a composite function, right? We can rewrite square root of 8 T + 16 as 8 T + 16 to the power of 1/2. And we're going to begin by applying the power rule for the outer function to get 1/2 multiplied by 8 T + 16. To the power of. 1/2, and according to the chain rule, we're multiplying by the derivative of 8 T + 16, and the derivative of -4 is 0, right? Because it is a constant. In the denominator, the derivative of T is a 1. So now let's go ahead and simplify. We can factor out 1/2 to get 1/2 multiplied by limit as T approaches 0 off. Now, for the first term, we are going to use the reciprocal rule that's 1 divided by square root of 8 T + 16. This is what -1/ represents as an exponent, and we are multiplying by the derivative of 8 T + 16, which is 8. And we can factor out 8, right? So that'd be 8 divided by 2, which is 4. Now, let's go ahead and perform the direct substitution once again. We got 4 multiplied by. 1 divided by square root of 8th multiplied by 0 + 16. This gives a square root of 16. So now 4 divided by a square root of 16 is equal to 1, which is our final answer for this problem. Thank you for watching.

Use l’Hôpital’s rule to find the limits in Exercises 7–52.37. lim (y → 0) (√(5y + 25) - 5) / y

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Use l’Hôpital’s rule to find the limits in Exercises 7–52.
37. lim (y → 0) (√(5y + 25) - 5) / y