More limits Evaluate the following limits.
lim_x→1 (x ln...

Question

More limits Evaluate the following limits.

lim_x→1 (x ln x - x + 1) / (xln²x)

Accepted Answer

Welcome back, everyone. In this problem, we want to evaluate the limit as X approaches 1 of 4 X minus 2 XLNX minus 4 divided by X 2 LN X. A says it's -2, B 2 C2, and D says it's 4. Now, let's first try to to evaluate the limit by direct substitution, OK? So what that means is that we are going to put X equals 1 into our expression. Now when we do that, OK. So that's just right, the expression out. Then we're going to get 4 multiplied by 1 minus 2 multiplied by 1 LN 1. Minus 4, OK, all divided by 1 squared multiplied by LN 1. Now, we know that the natural log of 1 is 0, OK? So this is gonna be 4 minus 0, because that's 2 multiplied by 0. So 4 minus 0 minus 4, which equals 0, and in our denominator, it's gonna be 1 squared multiplied by 0, which is 0. Notice here that the denominator is 0, so that means our function is undefined. In other words, our limit is in an indeterminate form. So to figure out the limit, we'll need to use Lapita's rule. What does that rule say? We recall that what it basically says is that if the limit as X approaches C of a quotient F of X divided by G of X. Is in an indeterminate form. OK. Then we can say that the limit as X approaches C. Of our caution. Is going to be equal to the limit as X approaches C of the derivatives of the numerator F of X and the denominator G of X respectively. So essentially, if we differentiate our numerator and denominator, then we can apply it to Lapital's rule. So let's go ahead and do that. And this works if the limit on the right exists, OK? So let's let F of X be equal to our numerator, 4 X minus 2 XLN X minus 4, OK. And let's let our denominator G of X be equal to X 2 LNX. In other words, let's let G of X be our denominator. In this case, that means then that the derivative of our numerator F of X will be equal to 4, OK, minus. Now remember we can differentiate 2 XLNX using the product rule, and that's going to give us 2 multiplied by LNX minus 2X multiplied by 1 divided by X, OK? And now when we simplify that, sorry, plus, this should be plus. I know when we simplify that, OK, we're gonna get uh 4 minus 2 LNX minus 2. Which we can simplify to be 4 minus 2 multiplied by LNX plus 1. OK? So this is the derivative of our numerator. On the other hand, for our denominator. Again, we can also apply the product rule for X2 LNX and that is gonna be LNX multiplied by 2 X plus X2 multiplied by 1 divided by X. Which is going to simplify to 2 XLNX plus X, OK? So now that we have the derivatives of the numerator and denominator for quotient, we can use Lapita's rule, which means that the limit as X approaches 1. Uh, 4 X minus 2 XL and X minus 4. Divided by X 2 LN X. Is going to be equal to the limit as X approaches 1. Of were -2 multiplied by LNX + 1. Divided by 2 XLNX plus X. Now, we can substitute X equals 1 into our limit. So this is going to be 4 minus 2 multiplied by the natural log of 1 + 1. All divided by 2 multiplied by 1 LN 1. Plus one. Now in this case here, we, like we said, we know that the natural log of 1 is 0, so this is going to be 4 minus 2 multiplied by 1. That's 4 minus 2, which is 2. And in our denominator 2 multiplied by LN 1 is 0, so that's gonna be 2 divided by 1 because the 0 was adding to 1 and 2 divided by 1 equals 2. Thus our limit evaluates to 2. If we look back at our answer choices, that tells us C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

More limits Evaluate the following limits.
lim_x→1 (x ln x - x + 1) / (xln²x)

Key Concepts

Limits

Indeterminate Forms

L'Hôpital's Rule

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