use the limit definition of the derivative to find the slope of the tangent line to the curve f(x)=6x² at x…

Explanation:

Step1: Recall limit - definition of derivative

The limit - definition of the derivative of a function $y = f(x)$ at $x=a$ is $f^{\prime}(a)=\lim_{h\rightarrow0}\frac{f(a + h)-f(a)}{h}$. Here, $f(x)=6x^{2}$ and $a = 2$. First, find $f(2 + h)$ and $f(2)$. $f(2 + h)=6(2 + h)^{2}=6(4 + 4h+h^{2})=24+24h + 6h^{2}$ and $f(2)=6\times2^{2}=24$.

Step2: Substitute into the limit - formula

$f^{\prime}(2)=\lim_{h\rightarrow0}\frac{f(2 + h)-f(2)}{h}=\lim_{h\rightarrow0}\frac{(24 + 24h+6h^{2})-24}{h}$. Simplify the numerator: $\frac{24 + 24h+6h^{2}-24}{h}=\frac{24h+6h^{2}}{h}$.

Step3: Simplify the fraction

$\frac{24h+6h^{2}}{h}=\lim_{h\rightarrow0}\frac{h(24 + 6h)}{h}$. Since $h\neq0$ (as we are taking the limit as $h$ approaches 0, not setting $h = 0$), we can cancel out the $h$ terms. So, $\lim_{h\rightarrow0}(24 + 6h)$.

Step4: Evaluate the limit

As $h\rightarrow0$, we substitute $h = 0$ into $24+6h$. Then $\lim_{h\rightarrow0}(24 + 6h)=24$.

Answer:

24