Find the limit, if it exists.
-$\lim _ { h \rightarrow 0 } \frac { ( 1 + h ) ^ { 1 / 3 } - 1 } { h }$
A) 0
B) Does not exist
C) $1 / 3$
D) 3

Question

Quizplus · Accepted Answer

We can use the limit definition of the derivative to evaluate this limit. Let $f(x) = x^{1/3}$, then
$$\lim_{h\rightarrow 0} \frac{(1+h)^{1/3}-1}{h} = \lim_{h\rightarrow 0}\frac{f(1+h)-f(1)}{h} = f'(1) = \frac{1}{3}$$
where the last equality follows from the power rule of differentiation. Therefore, the answer is (C).

Find the Limit, If It Exists lim⁡h→0(1+h)1/3−1h\lim _ { h \rightarrow 0 } \frac { ( 1 + h ) ^ { 1 / 3 } - 1 } { h }limh→0​h(1+h)1/3−1​

Find the Limit, If It Exists $\lim _ { h \rightarrow 0 } \frac { ( 1 + h ) ^ { 1 / 3 } - 1 } { h }$