use the definition mtan = lim x→a f(x) - f(a) / x - a to find the slope of the line tangent to the graph of…

Explanation:

Step1: Identify a and f(a)

Given (P(-3,-9)), so (a=-3) and (f(a)=f(-3)=-9). Also (f(x) = 5x + 6).

Step2: Substitute into the formula

[ \begin{align*} m_{\tan}&=\lim_{x\rightarrow - 3}\frac{f(x)-f(-3)}{x - (-3)}\ &=\lim_{x\rightarrow - 3}\frac{(5x + 6)-(-9)}{x+3}\ &=\lim_{x\rightarrow - 3}\frac{5x + 6 + 9}{x + 3}\ &=\lim_{x\rightarrow - 3}\frac{5x+15}{x + 3} \end{align*} ]

Step3: Simplify the expression

Factor out 5 from the numerator: (\lim_{x\rightarrow - 3}\frac{5(x + 3)}{x + 3}). Since (x\neq - 3) (in the limit - sense), we can cancel out (x + 3). So (m_{\tan}=\lim_{x\rightarrow - 3}5).

Step4: Evaluate the limit

The limit of a constant function (y = 5) as (x) approaches any value is 5. So (m_{\tan}=5).

Answer:

5