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Evaluate the limit, if it exists.

$ \displaystyle \lim_{h \to 0}\frac{(2 + h)^3 - 8}{h} $

$12$

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Missouri State University

Baylor University

University of Nottingham

We want to evaluate the limit of this expression as H approaches zero. Now if we try to directly substitute zero in for H uh we are going to get the indeterminant form of a limit. We're going to get 0/0. Uh So let's let's see if that's really the case plugging in zero for H If H 02 plus zero is to two Cubed would be eight. Okay so let's just write it down real quick. Okay if we just substitute zero in for age we would get two plus zero, cute minus eight over Once again plugging in zero for age. So this would be zero. Well two plus zero is too 2 to the 3rd is eight. Eight subtract 8 0. So we would have zero up top over zero. So directly substituting in zero for H uh brings us to the indeterminate form of a limit. 0/0. So basically we didn't get anywhere. So in order to actually calculate this limit, we are going to have to expand the numerator. We're actually going to uh do two plus H. Two the third and which means two plus eight times itself. Three times we're going to actually expand this and see if that helps us find the limit now. Two plus H. Two. The third is going to be a plus 12 H plus six H squared plus H. Cute. So two plus h. two. The third is this expression right here, we still have to write down to -8 and that whole thing. The whole expression gets put over each H and the denominator and let me fix this. Just supposed to be an L. Right here. So let's clean it up just a little bit. Okay so the limit of this expression that's H approaches zero is equal to the limit of this expression as H approaches zero. Now we can do a little bit of simplifying. We have eight here. Subtract eight there so those aides will cancel. And so we really had the limit of 12 H plus six. Eight square plus H cube. All divided by H. As H approaches zero. Each of these terms has an H. Uh has an H. So we are going to factor out the greatest common factor which is H. So rewriting this limit after we factor out the greatest common factor of H. Out of the numerator. Uh Well the numerator can be rewritten As aged times 12 plus six H. Plus each to the second and that is all over. H. You can distribute the multiplication by this age. To confirm that this is equal to what we had 12 times H. Six H. Times ages 68 squared eight square times ages. H. Cute. So the limit of this expression equals the limit of this expression now equals the limit of this expression. Well we can cancel out these ages times and by H. In the uh numerator. And dividing by agent. The denominator. Let's cancel out those H. Is So now we just have to take the limit of this expression as a jew approaches zero and we can take the limit of this expression. Uh As H approaches zero simply by plugging into zero everywhere you ch so 12 plus six times +06 times H will approach the limit of six times zero as H approaches zero plus a squared zero square. So now we are actually able to sub directly substitute zero in for H into this expression. Well uh six times 000 square to zero. And so our limit is 12. So the limit of this expression as H approaches zero is equal to 12.