consider the function f(x) = { -4x/(x - 3), x < 5; -7x + 25, x > 5. which type of discontinuity occurs at x…

Explanation:

Step1: Calculate left - hand limit

We find $\lim_{x\rightarrow5^{-}}f(x)$. For $x < 5$, $f(x)=-\frac{4x}{x - 3}$. Then $\lim_{x\rightarrow5^{-}}-\frac{4x}{x - 3}=-\frac{4\times5}{5 - 3}=-\frac{20}{2}=- 10$.

Step2: Calculate right - hand limit

We find $\lim_{x\rightarrow5^{+}}f(x)$. For $x>5$, $f(x)=-7x + 25$. Then $\lim_{x\rightarrow5^{+}}(-7x + 25)=-7\times5+25=-35 + 25=-10$.

Step3: Analyze discontinuity type

Since $\lim_{x\rightarrow5^{-}}f(x)=\lim_{x\rightarrow5^{+}}f(x)=-10$, but the function is not defined at $x = 5$, this is a removable discontinuity.

Answer:

removable