let f(x) = { (x² - 144)/(x + 12) if x < 0; (x² - 144)/(x - 12) if x > 0 }. find a) lim x→ - 12 f(x), b) lim…

let f(x) = { (x² - 144)/(x + 12) if x < 0; (x² - 144)/(x - 12) if x > 0 }. find a) lim x→ - 12 f(x), b) lim x→0 f(x), and c) lim x→12 f(x). a) select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. lim x→ - 12 f(x) = □ (simplify your answer.) b. the limit does not exist. b) select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. lim x→0 f(x) = □ (simplify your answer.) b. the limit does not exist. c) select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. lim x→12 f(x) = □ (simplify your answer.) b. the limit does not exist.

let f(x) = { (x² - 144)/(x + 12) if x < 0; (x² - 144)/(x - 12) if x > 0 }. find a) lim x→ - 12 f(x), b) lim x→0 f(x), and c) lim x→12 f(x). a) select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. lim x→ - 12 f(x) = □ (simplify your answer.) b. the limit does not exist. b) select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. lim x→0 f(x) = □ (simplify your answer.) b. the limit does not exist. c) select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. lim x→12 f(x) = □ (simplify your answer.) b. the limit does not exist.

Answer

Explanation:

Step1: Simplify the function for (x < 0)

Factor (x^{2}-144=(x + 12)(x - 12)). So for (x<0), (f(x)=\frac{x^{2}-144}{x + 12}=\frac{(x + 12)(x - 12)}{x+12}=x - 12) (for (x\neq - 12)).

Step2: Find (\lim_{x\rightarrow - 12}f(x))

Since for (x<0), (f(x)=x - 12), then (\lim_{x\rightarrow - 12^{-}}f(x)=-12-12=-24). The left - hand limit exists. So (\lim_{x\rightarrow - 12}f(x)=-24).

Step3: Analyze (\lim_{x\rightarrow0}f(x))

For (x\rightarrow0), we need to consider the left - hand limit ((x<0)) and right - hand limit ((x>0)). Left - hand limit ((x<0)): (f(x)=x - 12), (\lim_{x\rightarrow0^{-}}f(x)=0 - 12=-12). Right - hand limit ((x>0)): (f(x)=\frac{x^{2}-144}{x - 12}=\frac{(x + 12)(x - 12)}{x - 12}=x + 12) (for (x\neq12)), (\lim_{x\rightarrow0^{+}}f(x)=0 + 12=12). Since (\lim_{x\rightarrow0^{-}}f(x)\neq\lim_{x\rightarrow0^{+}}f(x)), the limit does not exist.

Step4: Analyze (\lim_{x\rightarrow12}f(x))

Left - hand limit ((x<0)) is not relevant as we approach from positive side for (x\rightarrow12). For (x>0), (f(x)=\frac{x^{2}-144}{x - 12}=x + 12) (for (x\neq12)), (\lim_{x\rightarrow12^{+}}f(x)=12 + 12=24). The left - hand limit (considering values close to 12 from the left side where (x>0)) also gives (x + 12) (since the function is (f(x)=\frac{x^{2}-144}{x - 12}) for (x>0)), (\lim_{x\rightarrow12^{-}}f(x)=12+12 = 24). So (\lim_{x\rightarrow12}f(x)=24).

Answer:

a) A. (\lim_{x\rightarrow - 12}f(x)=-24) b) B. The limit does not exist. c) A. (\lim_{x\rightarrow12}f(x)=24)