60–81. Limits Evaluate the following limits. Use l’Hôpital’s R...

Question

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.

lim_ x→0 ⁺ | ln x | ˣ

Accepted Answer

Evaluate the limit and apply the hopping tools rule if necessary. We have the limit as X approaches 0 from the right of natural log to X in the absolute value, raise to the 2 x power. We have 4 possible answers, being 0, infinity, 1, or does not exist. Now, we're gonna simplify our expression. You have natural log 2 X in absolute value, raised to the 2 X. If we were to follow this graph, we see the limit, as X approaches 0 to the right. Heads towards Negative infinity So then we can say that in this range, our natural log function is negative. We'll write this now. As negative, natural log 2X all race to the 2 X power. Now, to solve this, we will set this equal to Y and then take the natural log on both sides. Now we have natural luck. Of something raised to an exponential, we can make use of one of our log properties. We can move the exponent in front of our products. Natural log Y equals 2 X natural log of negative natural log to X. No. We need to use the hospital's role. Currently, we have the form. Of 0, multiplied. By infinity. So we need to rewrite this. We can write this by moving our 2 X into the denominator. And dividing 1 divided by 2 X. Now we can use Hato's rule. Which tells us that if we have Function Where the limits are in the form infinity divided by infinity, or 0 divided by 0, we can take the derivative of the numerator and the derivative of the denominator, and divide them. Now, if we take this derivative. We would get on the numerator of the function. 1 divided by negative natural log, 2 X. And because this is actually a chain rule derivative, we need the inside derivative of negative natural log to X. You can see the inside derivative. His 2 divided by 2 X. In our denominator, we have the derivative of 1 divided by 2 X. This would be -1 divided by 2 X squared. We can now simplify. X goes away, And the X squared loses one of its Xs. And our twos go away. We end up getting 1 divided by negative natural log 2 X multiplied by -2. All divided by -1 divided by X. We would then end up getting -2 X in the numerator by moving our X from the denominator up. Divided by negative natural log of 2 X. No. We can check our limit as it approaches 0 on the rights. So Limit. X approaches 0 from the right. We know that Native natural log to X. Approaches negative infinity. 2 X Approaches 0 We can notice That the 2 X term approaches zero way slowly, way more slow actually than the natural log function. So overall, this fraction should approach 0. So if we rewrite our function. We have Nao Y. Equals 0 Meaning that if we take E on both sides to get rid of the natural log, our limit should approach E to the 0 or 1. That means the answer to this problem. His answer C. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_ x→0 ⁺ | ln x | ˣ

Key Concepts

Limits

Natural Logarithm

l'Hôpital's Rule

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