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Numerade Educator

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Problem 17 Easy Difficulty

Evaluate the limit, if it exists.

$ \displaystyle \lim_{h \to 0}\frac{(-5 + h)^2 - 25}{h} $

Answer

-10

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Video Transcript

Okay, so we want to find a limit of this expression as H approaches zero. The first thing that we are going to do is do the negative five plus H squared using oil. So we're really taking the limit As a church approaches zero of while negative five plus H two. The second if you multiply it out negative five plus H times negative five plus H. You're going to get a positive 25 plus in H squared Uh -10. H. So negative five plus H squared is 25 plus H squared uh minus 10 H. And then we still have to attract this 25 And all that gets to by two by H. So the limit of this expression as H approaches zero is equal to the limit of this expression as H approaches zero, You can see we can do a little simplifying 25 -25 Cancell. And so we end up with the limit As a teacher approaches zero of eight squared minus 10 H over H Now I'm going to factor in h greatest common factor of H out of these two terms. So I'm going to rewrite the h squared minus 10 H as H times H minus 10 if you do eight times H minus 10 you'll get eight squared minus 2, 10. H. That is still being divided by H and factoring that uh works out because then I can cancel out these ages. Okay. Times in by H dividing by H that cancels. So I'm really just taking the limited as H approaches zero of H minus 10. Well, as H approaches zero, of course, this is going to approach zero, and so as a church approach zero, H minus 10 is going to approach zero minus 10, which would simply just be negative 10.