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_x→ 0⁺ (x - 3 √x) / (x - √x)

Accepted Answer

Welcome back, everyone. Find the limit using Lapito's rule if required. Limit as X approaches 0 from the right of 3 X minus 5 square root of X, all divided by 2X minus 3 square root of X. We're given 4 answer choices A says 5 divided by 3, B 3 divided by 2, C 2 divided by 3. In the 3 divided by 5. So let's rewrite the given limit, limit as X approaches 0 from the right of 3X minus 5 square root of X. All divided by 2 x minus 3 square root of X. Let's begin with the right substitution. We're going to get 3 multiplied by 0, which is 0, minus 5 multiplied by square root of 0, which is 0, and in the denominator we get 2 multiplied by 0, which is 0, minus 3 multiplied by 0, which is 0. So essentially, we get 0 divided by 0, which is an indeterminate form, and it works for Lapital's rule, right? So Lapto's rule says that if we get an indeterminate form such as 0 divided by 0 or infinity divided by infinity, we can simply differentiate the numerator, so 3 x minus 5 square root of x derivative, and we can differentiate the denominator. 2 x minus 3 square root of X, which will give us two new functions. And we will try and apply the right substitution once again until we get. A finite limit. So in this case, well, we have to identify each derivative. Let's recall that the derivative of square root of X is 1 divided by 2 square root of X. We will need that. So, the derivative of 3 X is 3 because the derivative of X is 1. We are subtracting 5 multiplied by the derivative of square root of X, which is 5 divided by 2 square root of X. And now in the denominator, we get 2 minus we divided by 2 square root of X. Now, what we're going to do is simply find the common denominator. For each part. So we get limit as X approaches 0. From the right The common denominator for the top is 2 square root of X, so we get 6 square root of X minus 5 divided by. 2 square ro of backs. Now, for the denominator, we are going to get 2 multiplied by 2 square root of X, that's 4 square root of X minus 3. All divided by 2 square root of X. Notice that each part is divided by the same denominator, so we can just cancel those out, and we get a ratio between 6 x minus 5 and 42 X minus 3. So let's apply the re substitution once again. We get 6 square root of 0 minus 5 divided by. 4 square root of 0 minus 3. Now, the first terms are zeros, right? So we can just say the first one is 04 square of 0 is 0, and we get -5 divided by -3, which is 5/3. And that means That the final answer to this problem corresponds to the answer to a 5/3. Thank you for watching.

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

lim_x→ 0⁺ (x - 3 √x) / (x - √x)

Key Concepts

Limits

L'Hôpital's Rule

Square Roots and Their Properties

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