 In this video, we are going to prove that the limit as h approaches zero of e to the h minus one over h is equal to one. We're going to see that this is a very important difference quotient necessary to compute the derivative of e to the x in the future. So the crux of this proof is based upon the fact that e is the limit as h approaches zero of the expression one plus h to the one over h power. So that's actually how we're going to define the number e. And so because of that, we're going to manipulate this limit statement to give us the number e to prove the limit in question right here. And this will be a consequence of the squeeze theorem. And so it turns out this expression that is, since e is equal to the limit of one plus h to the one over h, it's particularly the right-handed limit. It's also the left-handed limit. Now we're going to play around with the right-handed limit and the left-handed limit a little bit differently here. So if we investigate this limit a little bit more detail, we see that if h is just a teeny bit bigger than positive, if h is positive as it's a teeny bit bigger than zero, then it turns out that one plus h to the one over h is going to be less than or equal to e, right? And so as h gets closer and closer to zero from the positive side, this inequality becomes tighter and tighter and thus becomes equality when the limit is taken here. So this is true when h is close to zero. So working with this inequality then one plus h to the one over h power is less than or equal to e. If I take the h power of both sides, we see that these powers will cancel. We get one plus h is less than or equal to e to the h. For then, we subtract one from both sides. We get h is less than or equal to e to the h minus one. And then finally, if you divide both sides by h, you get this final statement right here. That one is less than or equal to e to the h minus one over h. If you've seen previous videos in our lecture series, the color coding scheme will hopefully make sense to you. The yellow function e to the h minus one over h, this is the function that we want to take the limit of. The orange function is going to be the lower bound of that function for forthcoming squeeze theorem application. So when h, when h is positive just a little bit bigger than zero, we get this inequality here. What is less than e to the h minus one over h? Well, since e, of course, is the limit, it's the limit as h approaches zero one plus h to the one over h. It's also the left-handed limit. So e equals the limit as h approaches zero from the left of one plus h to the one over h. Now, if you're approaching zero from the left, that means your number is negative. And so again, as we as we inspect this limit in a little bit more detail, we see that if h is a little bit smaller than zero that it's on the negative side, then e will be less than or equal to one plus h to the one over h power. Now, if h is positive, that means negative h is the negative. And so we then substitute this inequality into this expression right here to see the following. So if h is positive, negative h is negative. And we see that e will be less than or equal to one plus negative h to the one over negative h power, for which if you have one minus, one plus negative h such as one minus h, and if you have one over negative h such as negative one over h, which negative exponents mean that you take the reciprocal of the expression. So one minus h to the negative one over h power, that becomes one over one minus h to the one over h power. Let's play around with this inequality for a second then. So we have this side and this side. If we take the h power on the right, the powers will cancel. We do that on the left hand side as well. We end up with this expression e to the h is less than or equal to one over one minus h. Now, this is sort of the curious part. We're going to subtract one from both sides of this expression. So the left hand side becomes e to the h minus one. The right hand side becomes one over one minus h minus one, for which as we want to add together these fractions, we're going to take the number one and reproportionate. One will become one minus h over one minus h, for which now as there's a common denominator, we can add the numerators together. You're going to get one minus one, and then you're going to have a negative negative h, which a double negative becomes an h. And so we see that e to the h minus one is less than or equal to h over one minus h. For which in this situation, if we divide both sides by h, what we see is the left hand side will become e to the h minus one over h. The right hand side as these h's cancel, we get one over one minus h. Thus giving us the statement we see right here. And again, with our color coding scheme from before, the yellow function is the function that's going to get squeezed between two others. The green function is the function on the right. And so if we summarize our findings here, we see the following right here. When h is just a little bit bigger than zero on the positive side, we see that one will be less than or equal to e to the h minus one over h, which is less than or equal to one over one minus h. Now if we take the right-handed limits as h approaches zero from above of these expressions, well one is a constant function, therefore its limit will be one. On the other hand, if you take one over one minus h as h approaches zero, this will become one over one minus zero, which simplifies just to be one. So we see that this function approaches one, this function approaches one, and therefore the function e to the h minus one over h gets squeezed to that same limit that is by the by the squeeze theorem we see that the limit as h approaches zero from above, it can be e to the h minus one over h, that that limit will equal in fact one. Now this was under the assumption that h was greater than zero, so it's a positive number. By a similar argument, if we assume h is less than zero, it turns out all of these inequalities will get flipped around. You're probably used to the fact that if you times an inequality by negative one, it changes the directions. It's a little bit more involved in that, but replacing h with a negative, it will have the consequence that these inequalities get turned around. So when h is less than zero, we see that one over one minus h is less than or equal to e to the h minus one over h, which is less than or equal to one. But the limits are still the same as h approaches zero from the left. One will approach one and one minus one over one minus h will approach one as well. So the squeeze theorem applies again for which shows us that the left-handed limit is equal to zero. Now if the right-handed limit, sorry, the left-handed limit will go to one, excuse me. So if the right-handed limit gives us a one and the right-handed limit gives us one and the left-handed limit gives us one, then the limit, the two-sided limit, that is, will be their common value. And we see that the limit as h approaches zero of e to the h minus one over h is equal to one. And so this follows by an application of the squeeze theorem, but more importantly, it follows from the fact that e was the limit. We define e to be the limit of one plus h to the one over h power as h goes to zero.