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

Question

Indeterminate Powers and Products

Find the limits in Exercises 53–68.

55. lim (x → ∞) (ln x)^(1/x)

Accepted Answer

Welcome back everyone. Compute the limit as x approaches infinity of LN X. Raise the power of 1 divided by LN X. A says 0. B says 1, C says E, and D 1 divided by E. For this problem, let's rewrite the limit, limit as X approaches infinity. Of L N X raises the power of 1 divided by LN of X. Let's remember that the first step is to attempt direct substitution. So, if X approaches, I'm sorry, if X approaches infinity, LN of X also approaches infinity, and 1 divided by infinity is going to give us 0. So, we get an indeterminate form. Which is infinity, it's the power of zero, right? So what we're going to do is simply take the natural logarithm of our function by first of all, introducing L and L is going to be assigned to the limited self. Our goal is to find the value of L. If we take LN of L, we will be able to apply the properties of logarithms and get the value of L later on by using exponentiation. So LN of L is going to be the limit as X approaches infinity of LN of LN of X raises the power of 1 divided by LN of X. And now we can apply the properties of logarithms and basically bring down the exponent. Let's go ahead and see that. This is equal to limit as x approaches infinity of. LN of LN of X divided by because we're multiplying by 1 and dividing by LN of X. Now if we attempt direct substitution, we get a land of infinity which is infinity. A land of infinity returns as infinity still, and in the denominator we get a land of infinity, which is also infinity. For this indeterminate form, we can apply Lapito's rule. It says that we want to differentiate the numerator and the denominator. So we take LM of LMX derivative and divide by LMX derivative. Let's find each derivative in the numerator, we have a composite function. So, first of all, we differentiate natural log with respect to LN X, which gives us 1 divided by LN X. And according to the chain rule, we're multiplying by LNX derivative. And in the denominator, we have LNFX derivative which can now be canceled out, right? So now we got a limit as X approaches infinity. Of one divided by LN of X. Which is going to be equal to one divided by. As x approaches infinity, Allan of x approaches infinity. Remember 1 divided by infinity is equal to a limit of 0 because we have a constant divided by an infinitely large number. However, this is not the final answer yet. This is LN of L, so the limit itself L is going to be equal to E, raise to the power of 0, which is equal to 1. So the correct answer for this problem corresponds to the answer choice B. Thank you for watching.

Indeterminate Powers and ProductsFind the limits in Exercises 53–68.55. lim (x → ∞) (ln x)^(1/x)

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Indeterminate Powers and Products
Find the limits in Exercises 53–68.
55. lim (x → ∞) (ln x)^(1/x)