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Numerade Educator

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Problem 31 Easy Difficulty

Evaluate the limit, if it exists.

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

Answer

3$x^{2}$

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Video Transcript

So here were asked to evaluate specific limit. The limit as h approaches zero of X plus H cube minus X cubed divided by H. So let's first expand the numerator. So manly we're doing a binomial expansion here. So this would just be X cubed plus three X squared H plus three X H squared plus H cubed minus X cubed divided by H. So this is equivalent to the limit. As H approaches zero, the X keeps canceled out. And we also can factor out an H. In this case. So we have a church of three X squared Plus three x. H plus H squared divided by H. And we can cancel out the H is from the numerator and denominator. And we're left with the limit as h approaches zero of three x squared plus three X H plus H squared. We don't have any indeterminant form anymore. So we can directly evaluate so H zero and zero. So this would just be equivalent to three X squared. So this was giving the general limit form of the derivative of execute and we could also find the derivative of X cubed mainly by taking the derivative of X cubed which is just equivalent to three X squared, which agrees with this. Alternatively we can also apply lawful rule where we would take the derivative of the numerator and the derivative the denominator in order to find the limit. But that would be slightly more complicated in this case and we can just use the limit definition of the derivative to evaluate the derivative. And this is our final answer