Question

The limit represents the derivative of some function f at some number a. State such an f and a. lim h--->0 ((1+h)^7-1)/h

Answer 1

From the definition of the derivative:
\[f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}\]
You want to find a function \(f\) such that
\[\begin{cases}f(a+h)=\dfrac{1}{(1+h)^7}\\\\
f(a)=1\end{cases}\]
Hint:
\[f(a)=1=\frac{1}{1^7}\]

Answer 2

f(x) = x^6, a = 2
f(x) = x^7, a = 1
f(x) = x^8, a = 0
f(x) = x^7 − x, a = 1
f(x) = x^7 + x, a = 0 these are the answer choices and i know the third one is wrong but im guessing it could be the second one? @SithsAndGiggles

Answer 3

the fourth one is supposed to say f(x)=x^7-x, a=1

Answer 4

Oh I must have misread, you have
\[\begin{cases}f(a+h)=(1+h)^7\\
f(a)=1\end{cases}\]
And you have
\[f(a)=1=1^7\]
Suppose \(f(x)=x^7\). Then indeed, \(f(1+h)=(1+h)^7\), and if \(a=1\) then \(f(a)=1^7=1\).

Answer 5

thank you, i understand it now! @SithsAndGiggles