Question

for the following exercises, the given limit represents the derivative of a function ( y = f(x) ) at ( x = a ). find ( f(x) ) and ( a ).

1. ( lim_{h \to 0} \frac{(1 + h)^{2/3} - 1}{h} )

Explanation:

Step1: Recall the definition of the derivative

The derivative of a function \(y = f(x)\) 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 \(\lim_{h
ightarrow0}\frac{(1 + h)^{2/3}-1}{h}\).
By comparing with \(f^{\prime}(a)=\lim_{h
ightarrow0}\frac{f(a + h)-f(a)}{h}\), we can see that \(f(x)=x^{2/3}\) and \(a = 1\) since \(f(a+h)=(1 + h)^{2/3}\) and \(f(a)=1^{2/3}=1\).

Answer:

1. \(f(x)=x^{\frac{2}{3}}\)
2. \(a = 1\)