question\ngiven the function $g(x)=3x^{2}+7x$, which of the following is the correct limit definition of…

Explanation:

Step1: Recall the limit - definition of the derivative

The limit - definition of the derivative of a function $y = g(x)$ at $x = a$ is $g^{\prime}(a)=\lim_{h\rightarrow0}\frac{g(a + h)-g(a)}{h}$. Here, $a = 4$ and $g(x)=3x^{2}+7x$.

Step2: Find $g(4 + h)$

Substitute $x=4 + h$ into $g(x)$: [ \begin{align*} g(4 + h)&=3(4 + h)^{2}+7(4 + h)\ &=3(16 + 8h+h^{2})+28 + 7h\ &=48+24h + 3h^{2}+28 + 7h\ &=3h^{2}+31h + 76 \end{align*} ]

Step3: Find $g(4)$

Substitute $x = 4$ into $g(x)$: $g(4)=3\times4^{2}+7\times4=3\times16 + 28=48+28 = 76$.

Step4: Calculate $g(4 + h)-g(4)$

[ \begin{align*} g(4 + h)-g(4)&=(3h^{2}+31h + 76)-76\ &=3h^{2}+31h \end{align*} ]

Step5: Write the limit - definition of $g^{\prime}(4)$

[ \begin{align*} g^{\prime}(4)&=\lim_{h\rightarrow0}\frac{g(4 + h)-g(4)}{h}\ &=\lim_{h\rightarrow0}\frac{3(4 + h)^{2}+7(4 + h)-76}{h}\ &=\lim_{h\rightarrow0}\frac{3(4 + h)^{2}+7h+28 - 76}{h}\ &=\lim_{h\rightarrow0}\frac{3(4 + h)^{2}+7h - 48}{h} \end{align*} ]

Answer:

$\lim_{h\rightarrow0}\frac{3(4 + h)^{2}+7h - 48}{h}$ (the last option)