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.

35. lim (x → 0⁺) ln(x² + 2x) / ln x

Accepted Answer

Welcome back, everyone. Compute the following limit using Lapital's rule. Limit as x approaches 0 from the right of LN X2 + 4 X divided by LN of X A 1 B 0 C -1. Indeed, the limit does not exist. Let's rewrite the limit. We have limit as x approaches 0 from the right. Of LN of X2 + 4 X? Divided by a land of X. Our first step is to replace every X with 0. So we get LN of 0 squad + 4 multiplied by 0. That's LN of 0, right? And specifically, we're going to get LN of 0 from the right. Divided by LN of 0 from the right. Now, LN of 0 from the right there's going to be negative infinity. So we get negative infinity in the numerator. And in the denominator, we also get negative infinity, which is basically infinity divided by infinity. In this indeterminate form is acceptable. To apply Lapito's rule immediately. According to Lapito's rule, we want to differentiate the numerator and the denominator. So we take the derivative of LN X2 + 4 X and divide by the derivative of LN X. We can begin with the simpler one. The derivative of LN X is basically one divided by X. It is an elementary derivative. And now for the derivative of LN of X2 + 4 X, well, this is a composite function, so we want to apply the chain rule. First of all, we differentiate natural log with respect to the inner function to get one divided by. X2 + 4 X. And we multiply by the derivative of the inner function, so we multiply by the derivative of X2 + 4 X. The derivative of X2 + 4 X can be found using the power rule. And we're going to get 2 X + 4, right? So now, let's go ahead and simplify. This is going to be limit as X approaches 0 from the right. We can perform factorization to get 2 inferences X + 2. Divided by x in parent X plus 4, and we are dividing by 1 divided by X. Notice that in each case we have X in the denominator, so we can cancel it out. And we can simplify the resultant limit as follows. We are going to get. Limit as X approaches 0 from the right. We also have a constant, we can factor it out, so that'll be 2 limit as X approaches 0 from the right of. X + 2. Divided by X + 4. And now let's substitute X equals 0, so we're going to get 2 multiplied by. 0 + 2. Divided by 0 + 4. Which is now a finite value, right? We get 2 multiplied by 2 divided by 4, and that is equal to 1. Meaning the correct answer for this problem corresponds to the answer choice a. Thank you for watching.

Use l’Hôpital’s rule to find the limits in Exercises 7–52.35. lim (x → 0⁺) ln(x² + 2x) / ln x

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Use l’Hôpital’s rule to find the limits in Exercises 7–52.
35. lim (x → 0⁺) ln(x² + 2x) / ln x