Find the limit, if it exists.
-$\lim _ { h \rightarrow 0 } \frac { ( x + h ) ^ { 3 } - x ^ { 3 } } { h }$
A) Does not exist
B) $3 x ^ { 2 } + 3 x h + h ^ { 2 }$
C) $3 x ^ { 2 }$
D) 0

Question

Quizplus · Accepted Answer

Note that:
\begin{align*}
\lim_{hightarrow 0}\frac{(x+h)^3-x^3}{h}&=\lim_{hightarrow 0}\frac{(x+h-x)((x+h)^2+x(x+h)+x^2)}{h}\
&=\lim_{hightarrow 0}\frac{h(3x^2+3xh+h^2)}{h}\
&=\lim_{hightarrow 0}(3x^2+3xh+h^2)\
&=3x^2
\end{align*}
Thus, the answer is (C).

Find the Limit, If It Exists lim⁡h→0(x+h)3−x3h\lim _ { h \rightarrow 0 } \frac { ( x + h ) ^ { 3 } - x ^ { 3 } } { h }limh→0​h(x+h)3−x3​

Find the Limit, If It Exists $\lim _ { h \rightarrow 0 } \frac { ( x + h ) ^ { 3 } - x ^ { 3 } } { h }$