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→1⁻ (x/(x-1) - x/(ln x)

Accepted Answer

Hello there. Today we're gonna 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 1 from the right of X divided by the natural log of X minus 1 divided by X minus 1. Awesome. So it appears for this particular problem, we're asked to evaluate the specific limit, and we're also given a hint to use Lahapital's rule if necessary. So now that we know that we're ultimately trying to evaluate this limit, let's read off our multiple choice answers to see what our final answer might be. A is 1/3, B is 2/3. C is 3 halves, and D is 116 or 11 divided by 6. Awesome. So, first off, in order to evaluate the limit, and once again we're told that our limit is as X approaches 1 from the right, that's what that positive symbol in the power means. So, as X approaches 1 from the right of X divided by the natural log of X minus 1 divided by X minus 1, we should first try to combine the terms into a single fraction. So when we do that, we will get that X divided by the natural log of X minus 1 divided by X minus 1. is going to be equal to X multiplied by X minus 1 minus the natural log of X, fantastic, and then this will be all divided by Drum roll, the natural log of X multiplied by X minus 1. Fantastic. So now let us analyze the behavior of the numerator and denominator as X approaches 1 from the right. So we need to note that as X approaches 1 from the right, that will mean that X multiplied by X minus 1 will approach 0, and that the natural log of X will approach 0, so the numerator will tend to 0 minus 0, which is equal to 0. Fantastic. And the denominator, which is the natural log of X, so when you take the, now we're focusing on the denominator. So the denominator, the natural log of X multiplied by X minus 1, will also 10 to 0. Fantastic. So this will result in an indeterminate form, which, as we should recall, an indeterminate form, and let's make a note of this in red. So an indeterminate form is when we get an answer of 0 divided by 0 or infinity divided by infinity. So when we get an indeterminate form when we use a direct substitution method, We will be able to note that we can use Laja Patal's rule in order to solve, or rather to evaluate the limit. So because we got this indeterminate form, we can apply Lajapatal's rule, which will take, which will, as we should recall, which will state that the limit. Which was we should recall La Hopital's rule states that the limit, as X approaches A of F of X divided by G of X. is going to be equal to the limit as X approaches A F X divided by G of X. So it states that Lajapato's rule states that as the limit of the limit as X approaches A of the function 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 and divided by the first derivative of G of X. Awesome. So that means that this provided that, so this will be true provided that the limit on the right exists. So with that said, we need to now differentiate the numerator and denominator. So focusing our attention first on the numerator, we will determine that F of X is equal to X multiplied by X minus 1 minus the natural log of X. So that will mean that When we do our first derivative, that will mean that F of X, where this prime symbol denotes the first derivative, that will mean that F of X is equal to 1 multiplied by X minus 1. Plus X multiplied by 1, minus 1 divided by X, it's gonna be all equal to 2 X minus 1 minus 1 divided by X. Fantastic. So, with that said, our next step now is we now need to differentiate the denominator, which we're gonna be using for G of X. So G of X is equal to the natural log of X multiplied by X minus 1. And in order to perform this particular differentiation, we must recall and use the product rule. So when we do that, we will find that G of X is equal to 1 divided by X, multiplied by X minus 1 plus the natural log of X multiplied by 1, which will be all equal to X minus 1 divided by X plus the natural log of X. Fantastic. So therefore, our limit will become the limit as X approaches one from the right of. 2 X minus 1 minus 1 divided by X, divided by X minus 1 divided by X plus the natural log of X. Awesome. So, we need to note that as X approaches one from the right, both the numerator and the denominator will still tend to 0. So as we could see when we take 2 multiplied by 1 minus 1 will be equal to 1, so actually, right. So if we take 2 multiplied by 1 minus 1, and then don't forget that we also have to subtract 1 divided by 1, so that will mean it's equal to 0 because once again we have to plug in a value of 1 in for experts using the direct substitution method. So that's for our numerator, which also say N for numerator and then D for denominator. You will see that 1 minus 1 divided by 1 plus the natural log of 1 will be equal to, once again, 0. So once again, we have to apply Lahapatal's rule again to our new limit value that we already had differentiated one time. So we need to take the limit. So we need to take our limit of as X approaches 1 from the right of 2, X minus 1, minus 1 divided by X divided by X minus 1 divided by X plus the natural log of X, and we need to differentiate the numerator and denominator again. So with that said, differentiating the numerator, we will have 2 X minus 1 minus 1 divided by X. So when we take the derivative, we will find that it's 2 minus -1. So 2 minus -1 multiplied by X to the power of -2, which will be equal to 2 + 1 divided by X to the power of 2. Fantastic. And then if we take the Derivative once again of the denominator, which for this case we're going to be taking X minus 1 divided by X plus the natural log of X. And when we perform that derivative, we will get -1 multiplied by -1 multiplied by X to the power of -2, plus 1 divided by X, which will be all equal to 1 divided by X to the power of 2, plus 1 divided by X. Which will be equal to 1 + X, so 1 + X divided by X to the power of 2. Awesome. So now we could write our new limit. So our new limit is going to be the limit as X approaches 1 from the right of 2 + 1 divided by X to the power of 2, divided by 1 + X divided by X to the power of 2. And this will simplify to the limit as X approaches 1 from the right of 2 + 1 divided by X2. Divide, so actually, instead of saying 1, so we're taking the the limit as X approaches one from the right of 2 plus 1 divided by X2, once again we can simplify and instead we could write that this limit as X approaches one from the right of. 2 multiplied by X2. So 2 X2 + 1 divided by 1 + X. Fantastic because you can cancel out these X squares. So awesome. So now that we simplified, we now need to let X approach 1 from the right. So the numerator, focusing on her numerator now, that will mean that it's going to be 2 multiplied by 12 + 1, which will be equal to 3, and then similarly for the denominator, we'll take 1 + 1, which is going to be equal to 2. So that will mean that our limit is going to be equal to 3 divided by 2, and that is our final answer. All right, we did it. And once again, what we're doing is we're doing direct substitution. So since we're told that our limit is X is approaching 1 from the right, we plug in a value of 1 in for X. So when we do that, we will find that 2 multiplied by 1 to the power of 2 is going to be 2 and 2 + 1 is 3. And then 1 plus 1 will be 2, so our final answer is going to be 3 halves or 3 divided by 2, and this corresponds with multiple choice answer letter C, which once again is 3 halves or 3 divided by 2. 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→1⁻ (x/(x-1) - x/(ln x)

Key Concepts

Limits

l'Hôpital's Rule

Natural Logarithm (ln)

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