For what function f and number a is the
limit lim h to 0 (((32+h)^(1/5))-2)/h the value f'(a)?

Question

For what function f and number a is the
limit lim h to 0  (((32+h)^(1/5))-2)/h the value f'(a)?

anonymous · Answer

I assume you intend the following:$$\lim_{h\rightarrow 0}\dfrac{(32+h)^{1/5}-2}{h}$$Recall the limit definition of f'(a):$$\lim_{h\rightarrow 0}\dfrac{f'(a+h)-f'(a)}{h}$$Knowing the first term in the numerator of the given limit is f'(a+h), we can guess and say$$f'(a)=(32)^{1/5}=2\text{ } \checkmark$$Since this matches what is supposed to be f(a) from the original limit, we thus determine that a, the original input, is 32. For more solutions to textbook problems, be sure to check out http://www.slader.com/s/eWFrZXlnbGVl. I submit a lot of content there as well.

josee · Answer

what does f(x)=?

blacksteel · Answer

One slight correction to yakeyglee's answer:
 It should be
f'(a) = $$\lim_{h \rightarrow 0} [f(a + h) - f(a)]/h$$

Otherwise his answer is right.$$\lim_{h \rightarrow 0} [(32 + h)^{1/5} - 2]/h =$$
$$\lim_{h \rightarrow 0} [(32 + h)^{1/5} - (32)^{1/5}]/h$$

So a =32 and f(x) = x^{1/5}