Question

a function f is said to have a removable discontinuity at a if: 1. f is either not defined or not continuous at a. 2. f(a) could either be defined or redefined so that the new function is continuous at a. let f(x) = (2x² + 4x - 30)/(x - 3). show that f has a removable discontinuity at 3 and determine the value for f(3) that would make f continuous at 3. need to redefine f(3) = □.

Explanation:

Step1: Factor the numerator

Factor $2x^{2}+4x - 30$. We have $2x^{2}+4x - 30=2(x^{2}+2x - 15)=2(x + 5)(x - 3)$. So $f(x)=\frac{2(x + 5)(x - 3)}{x - 3}$.

Step2: Simplify the function

For $x
eq3$, we can cancel out the $(x - 3)$ terms. Then $f(x)=2(x + 5)$ for $x
eq3$.

Step3: Find the limit as x approaches 3

We find $\lim_{x
ightarrow3}f(x)$. Substitute $x = 3$ into $y = 2(x + 5)$. $\lim_{x
ightarrow3}f(x)=2(3 + 5)=16$.

Answer:

16