Evaluate the limit, if it exists. $\lim _{h \rightarrow 0} \frac{(x-h)^{7}-x^{7}}{h}$
A) $7 x^{6}$
B) $-7 x^{6}$
C) $-7$
D) 1
E) does not exist

Question

Quizplus · Accepted Answer

We can expand the numerator using the binomial theorem:
$$(x-h)^{7}=x^{7}-\binom{7}{1}x^{6} h+\binom{7}{2}x^{5} h^{2}-\cdots-\binom{7}{6}x h^{6}+h^{7}$$
So, 
$$\frac{(x-h)^{7}-x^{7}}{h}=-\binom{7}{1}x^{6}+\binom{7}{2}x^{5} h-\cdots-\binom{7}{6}x h^{5}+h^{6}.$$
As $h$ approaches $0$, all of the terms with $h$ in them vanish, leaving only $-\binom{7}{1}x^{6} = -7x^6$. Hence, the limit is $-7x^6$.

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Evaluate the Limit, If It Exists $\lim _{h \rightarrow 0} \frac{(x-h)^{7}-x^{7}}{h}$