Question

the limit represents the derivative of some function ( f ) at some number ( a ). state such an ( f ) and ( a ).
lim _{h
ightarrow 0} \frac{(1+h)^{4}-1}{h}
( \bigcirc f(x)=x^{3}, a=2 )
( \bigcirc f(x)=x^{4}-x, a=1 )
( \bigcirc f(x)=x^{4}, a=1 )
( \bigcirc f(x)=x^{4}+x, a=0 )
( \bigcirc f(x)=x^{5}, a=0 )

Explanation:

Step1: Recall the definition of the derivative

The derivative of a function \(f(x)\) at \(x = a\) is given by \(f^{\prime}(a)=\lim_{h
ightarrow0}\frac{f(a + h)-f(a)}{h}\).

Step2: Compare with the given limit

Given \(\lim_{h
ightarrow0}\frac{(1 + h)^{4}-1}{h}\).
If we set \(f(x)=x^{4}\) and \(a = 1\), then \(f(a + h)=(1 + h)^{4}\) and \(f(a)=1^{4}=1\).

Answer:

\(f(x)=x^{4}\), \(a = 1\) (the third option).