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anonymous
 one year ago
Evaluate the limit, if it exists. (If an answer does not exist, enter DNE.)
lim (x + h)^3 − x^3 (over) h
h → 0
anonymous
 one year ago
Evaluate the limit, if it exists. (If an answer does not exist, enter DNE.) lim (x + h)^3 − x^3 (over) h h → 0

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SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.0\(\large\color{slate}{\displaystyle\lim_{h \rightarrow ~0}\frac{(x+h)^3x^3}{h}}\) (Or the derivative of x\(^3\), for which you should get 3x\(^2\), by the power rule, thus we know what value must your limit be equal to. > If you have ever learned the power rule)

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.0You have to expand the \((x+h)^3\), at first.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0\[h^3+3h^2x+3hx^2+x^3\]

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.0Yes, and now, write that on top of your fraction instead of (x+h)³.

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.0\(\large\color{slate}{\displaystyle\lim_{h \rightarrow ~0}\frac{(x+h)^3x^3}{h}}\) \(\large\color{slate}{\displaystyle\lim_{h \rightarrow ~0}\frac{h^3+3h^2x+3hx^2+x^3x^3}{h}}\)

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.0\(x^3\) goes away, and then h will cancel.

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.0(The validity of dividing by h on top and bottom, as h\(\rightarrow\)0, is justified by the fact that you are taking values that are not actually equal to 0, rather near 0.)

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.0\(\large\color{black}{\displaystyle\lim_{h \rightarrow ~0}\frac{h^3+3h^2x+3hx^2\cancel{~+x^3x^3~}}{h}}\) \(\large\color{black}{\displaystyle\lim_{h \rightarrow ~0}\frac{h^3+3h^2x+3hx^2}{h}}\)
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