Evaluate the following limits. Use l’Hôpital’s Rule when it is...

Question

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→ e (ln x - 1) / (x - 1)

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Evaluate the limit. Use Lahapatal's rule if necessary. The limit as X approaches E of E minus X all divided by 3 minus. Natural log of x to the power of 3. Awesome. So it appears our final answer that we're ultimately trying to solve for is we're asked to evaluate this particular limit, and once we evaluated the limit, that will be our final answer we're ultimately trying to solve for. So with that in mind, let's read off our multiple choice answers to see what our final answer might be. A is 0, B is E divided by 6, C is E divided by 3, and D is 3 multiplied by E or 3 E. Awesome. So with that said, first off, in order to evaluate the limit, which our limit once again is the limit, as X approaches E of E minus X, all divided by 3 minus the natural log of X to the power of 3. We must first we need to first in order before we do anything, since we're asked to use La Hait's rule, we have to evaluate the sea limit using the direct substitution method to see if it will yield an indeterminate form. And if, if it yields an indeterminate form, only then can we apply this law hapas rule. So with that said, we need to focus on this X approaches E. And we need to plug in this value of E in for X. So, when we substitute in, and once again, let's make a note that we're substituting in the value that X equals E, so X equals E. So when we substitute in the value that X equals Z, the numerator, Which is, once again, so it be E minus X, it will become 0 because E minus E will equal to 0, and the denominator, which is 3 minus the natural log of X to the power of 3, it will become, once again, 3 minus the natural log of E to the power of 3. And once again, this will give us a value of, when we simplify. It'll give us 3 minus 3 multiplied by 1, which will give us a value of 0. So that means when we use this direct substitution method, we'll get 0 divided by 0 as an answer, which will mean that this is an indeterminate form. So, as you should recall, 0 divided by 0 is an indeterminate form, or in some cases you will get an indeterminate form where it's infinity divided by infinity. Awesome. So in order to use Lahapaal's rule, you use direct substitution, and you have to make sure it's when you plug in using direct substitution, you get an indeterminant form of 0 divided by 0 or infinity divided by infinity, and if you do get an indeterminate form, then you're allowed to use Lahapatal's rule. So now that we've confirmed that we can use Lahapatal's rule. So since we determine this, we can therefore once again apply Lahapaal's rule, and La Haital's rule states that if the limit is in the form of 0 divided by 0 or infinity to divided by infinity, then we're able to safely state that the limit as X approaches, A, we'll say that x approaches. A of F of X divided by G of X is equal to the limit as X approaches A F X divided by G of X. Fantastic. So, with that said, we need to note that this is provided that the limit on the right exists. So to quickly recap here, so essentially, Lahapato's rule states that the limit as X approaches A of F of X divided by G of X is equal to the limit as X approaches A of the first derivative of F of X divided by the first derivative of G of X. That's what that little prime symbol means. This prime symbol means it's the first derivative. Fantastic. So now what we need to do is we need to find the derivatives of the numerator and denominator. So we need to note and recall that the derivative of the numerator, which note that our numerator is E minus X, and the first derivative will be -1, and the derivative of the denominator, which once again numerator is the top expression and denominator is the bottom expression. Awesome. So once again, our denominator is 3 minus the natural log of X to the power of 3. And in order to perform this particular differentiation, we need to recall and use the chain rule. So we need to take D by DX of which D by DX is a fancy way of saying we're taking the derivative with respect to X. So we need to take D by DX of 3 minus the natural log of X to the power of 3. is equal to D by DX of negative natural log of X to the power of 3, which will be equal to negative D by DX of the natural log of X to the power of 3. Fantastic. So, We need to note that since the natural log of X to the power of 3, and let's make a note of this up above here in blue, let us note that the natural log of X to the power of 3 is equal to 3 multiplied by the natural log of X. So that will mean that the derivative will be D by DX of 3 multiplied by the natural log. Of X is equal to 3 divided by X. Fantastic. So 3 divided by X. So therefore, the derivative of the denominator, which I'll say that our final derivative for the denominator is going to be -3 divided by X because don't forget we pulled out this negative symbol out to the front of our derivative. So now we can officially apply Lahapatal's rules, so we can go ahead and write that the limit as X approaches E of E. Minus X all divided by 3 minus the natural log of X to the power of 3 is equal to the limit, as X approaches E of -1. So when you take with the limit of -1 divided by -3 divided by X, and when we simplify, this will be all equal to the limit as X approaches E of X divided by 3. So now we can now we need to use the direct substitution method again, so we need to substitute in the value of X equals E into our limit. And evaluate the expression. And when we do that, we will find that the limit as X approaches E. Of E divided by 3. So that means that our final answer is going to be E divided by 3, and that is our final answer. Hooray, we did it. So therefore, without a doubt, our final answer for the limit will be multiple choice answer letter C, which is E divided by 3. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ e (ln x - 1) / (x - 1)

Key Concepts

Limits

l'Hôpital's Rule

Natural Logarithm

Watch next