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

ightarrow0}\frac{(1 + h)^{\frac{2}{3}}-1}{h} )

1. ( lim_{h

ightarrow0}\frac{3(2 + h)^{2}+2-14}{h} )

1. ( lim_{h

ightarrow0}\frac{cos(pi + h)+1}{h} )

1. ( lim_{h

ightarrow0}\frac{(2 + h)^{4}-16}{h} )

1. ( lim_{h

ightarrow0}\frac{2(3 + h)^{2}-(3 + h)-15}{h} )

1. ( lim_{h

ightarrow0}\frac{e^{h}-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: Identify \(f(x)\) and \(a\)

We are given \(\lim_{h
ightarrow0}\frac{[2(3 + h)^{2}-(3 + h)]-15}{h}\)

First, find \(f(a)\):
When \(h = 0\), the expression inside the limit is \(\frac{[2(3+0)^{2}-(3 + 0)]-15}{0}\) (in the indeterminate form \(\frac{0}{0}\)).

If we set \(f(x)=2x^{2}-x\) and \(a = 3\)

Then \(f(a+h)=2(3 + h)^{2}-(3 + h)\) and \(f(a)=2\times3^{2}-3=18 - 3=15\)

Answer:

\(f(x)=2x^{2}-x\) and \(a = 3\)