L'Hôpital's rule: challenging problem

We want to figure out the limit as x approaches 1 of the expression x over x minus 1 minus 1 over the natural log of x. So let's just see what happens when we just try to plug in the 1. What happens if we evaluate this expression at 1? Well then, we're going to get a one here, over 1 minus 1. So we're going to get something like a 1 over a 0, minus 1 over, and what's the natural log of 1? e to the what power is equal to one? Well, anything to the zeroth power is equal to 1, so e to the zeroth power is going to be equal to 1, so the natural log of 1 is going to be 0. So we get the strange, undefined 1 over 0 minus 1 over 0. It's this bizarre-looking undefined form. But it's not the indeterminate type of form that we looked for in l'Hopital's rule. We're not getting a 0 over a 0, we're not getting an infinity over an infinity. So you might just say, hey, OK, this is a non-l'Hopital's rule problem. We're going to have to figure out this limit some other way. And I would say, well don't give up just yet! Maybe we can manipulate this algebraically somehow so that it will give us the l'Hopital indeterminate form, and then we can just apply the rule. And to do that, let's just see, what happens if we add these two expressions? So if we add them, so this expression, if we add it, it will be, well, the common denominator is going to be x minus 1 times the natural log of x. I just multiplied the denominators. And then the numerator is going to be, well, if I multiply essentially this whole term by natural log of x, so it's going to be x natural log of x, and then this whole term I'm going to multiply by x minus one. So minus x minus 1. And you could break it apart and see that this expression and this expression are the same thing. This right here, that right there, is the same thing as x over x minus 1, because the natural log of x's cancel out. Let me get rid of that. And then this right here is the same thing as 1 over natural log of x, because the x minus 1's cancel out. So hopefully you realize, all I did is I added these two expressions. So given that, let's see what happens if I take the limit as x approaches 1 of this thing. Because these are the same thing. Do we get anything more interesting? So what do we have here? We have one times the natural log of 1. The natural log of 1 is 0, so we have 0 here, so that is a 0. Minus 1 minus 0, so that's going to be another 0, minus 0. So we get a 0 in the numerator. And in the denominator we get a 1 minus 1, which is 0, times the natural log of 1, which is 0, so 0 times 0, that is 0. And there you have it. We have indeterminate form that we need for l'Hopital's rule, assuming that if we take the derivative of that, and put it over the derivative of that, that that limit exists. So let's try to do it. So this is going to be equal to, if the limit exists, this is going to be equal to the limit as x approaches 1. And let's take the derivative in magenta, I'll take the derivative of this numerator right over here. And for this first term, just do the product rule. Derivative of x is one, and then so 1 times the natural log of x, the derivative of the first term times the second term. And then we're going to have plus the derivative of the second term plus 1 over x times the first term. It's just the product rule. So 1 over x times x, we're going to see, that's just 1, and then we have minus the derivative of x minus 1. Well, the derivative of x minus 1 is just 1, so it's just going to be minus 1. And then, all of that is over the derivative of this thing. So let's take the derivative of that, over here. So the derivative of the first term, of x minus 1, is just 1. Multiply that times the second term, you get natural log of x. And then plus the derivative of the second term, derivative of natural log of x is one over x, times x minus 1. I think we can simplify this a little bit. This 1 over x times x, that's a 1. We're going to subtract one from it. So these cancel out, right there. And so this whole expression can be rewritten as the limit as approaches 1, the numerator is just natural log of x, do that in magenta, and the denominator is the natural log of x plus x minus 1 over x So let's try to evaluate this limit here. So if we take x approaches one of natural log of x, that will give us a, well, natural log of 1 is 0. And over here, we get natural log of 1, which is 0. And then plus 1 minus 1 over plus 1 minus 1 over 1, well, that's just going to be another 0. 1 minus 1 is zero. So you're going to have 0 plus 0. So you're going to get a 0 over 0 again. 0 over 0. So once again, let's apply l'Hopital's rule again. Let's take the derivative of that, put it over the derivative of that. So this, if we're ever going to get to a limit, is going to be equal to the limit as x approaches 1 of the derivative of the numerator, 1 over x, right, the derivative of ln of x is 1/x, over the derivative of the denominator. And what's that? Well, derivative of natural log of x is 1 over x plus derivative of x minus 1 over x. You could view it this way, as 1 over x times x minus 1. Well, derivative of x to the negative 1, we'll take the derivative of the first one times the second thing, and then the derivative of the second thing times the first thing. So the derivative of the first term, x to the negative 1, is negative x to the negative 2 times the second term, times x minus 1, plus the derivative of the second term, which is just 1 times the first term, plus 1 over x. So this is going to be equal to, I just had a random thing pop up on my computer. Sorry for that little sound, if you heard it. But where was I? Oh, let's just simplify this over here. We were doing our l'Hopital's rule. So this is going to be equal to, let me, this is going to be equal to, if we evaluate x as equal to 1, the numerator is just 1/1, which is just 1. So we're definitely not going to have an indeterminate or at least a 0/0 form anymore. And the denominator is going to be, if you evaluate it at 1, this is 1/1, which is 1, plus negative 1 to the negative 2. So, or you say, 1 to the negative 2 is just 1, it's just a negative one. But then you multiply that times 1 minus 1, which is 0, so this whole term's going to cancel out. And you have a plus another 1 over 1. So plus 1 And so this is going to be equal to 1/2. And there you have it. Using L'Hopital's rule and a couple of steps, we solved something that at least initially didn't look like it was 0/0. We just added the 2 terms, got 0/0, took derivatives of the numerators and the denominators 2 times in a row to eventually get our limit.