Indeterminate Powers and Products
Find the limits in Exe...

Question

Indeterminate Powers and Products

Find the limits in Exercises 53–68.

57. lim (x → 0⁺) x^(-1/ln x)

Accepted Answer

Welcome back, everyone. Evaluate the limit as x approaches 0 from the right of X to the power of 2 divided by LN of X. A says E 2 B 1 divided by equad C E and D 1 divided by E. Let's begin by rewriting the limit. The limit as X approaches 0 from the right. Of X is the power of 2 divided by LN of X. We're going to attempt direct substitution to get 0 from the right, raise the power of 2 divided by L and of 0 from the right. Now, let's simplify. This is going to be equal to. raises the power of 2 divided by negative infinity which basically approaches 0 and 0 to the power of 0 is an indeterminate form. For this indeterminate form, what we're going to do is simply assign L to our limit, and we're going to take LN of L to apply the properties of logarithms. So let's take a land of the limit. And we're going to get a limit as X approaches 0 from the right of LN of our function X to the power of 2 divided by LN of X. Which allows us to apply the power rule, we can bring down the exponent, and we're going to get a limit as X approaches 0 from the right of. 2 divided by Alan of X multiplied by Ellen of X. So now, in this case, notice that we can cancel out LN of X and we basically get The limit of a constant limit of 2, which is equal to that constant itself because it's independent of X. Remember that this is LN of the limit. So the limit L itself is the NI log. We're going to take E to the power of 2, which is Esquared. Meaning the correct answer for this problem corresponds to the answer choice a e squared. Thank you for watching.

Indeterminate Powers and ProductsFind the limits in Exercises 53–68.57. lim (x → 0⁺) x^(-1/ln x)

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Indeterminate Powers and Products
Find the limits in Exercises 53–68.
57. lim (x → 0⁺) x^(-1/ln x)