for the function shown below, find (if the quantity exists) (a) lim f(x), (b) lim f(x), (c) lim f(x), and…

for the function shown below, find (if the quantity exists) (a) lim f(x), (b) lim f(x), (c) lim f(x), and (d) f(0). x→0+ x→0− x→0 f(x)={15 - x², for x ≤ 0; 15 + x², for x > 0} (a) select the correct choice below and fill in any answer boxes in your choice. a. lim f(x)=□ x→0+ b. the limit does not exist. (b) select the correct choice below and fill in any answer boxes in your choice. a. lim f(x)=□ x→0− b. the limit does not exist. (c) select the correct choice below and fill in any answer boxes in your choice. a. lim f(x)=□ x→0 b. the limit does not exist. (d) select the correct choice below and fill in any answer boxes in your choice. a. f(0)=□ b. the value does not exist.

for the function shown below, find (if the quantity exists) (a) lim f(x), (b) lim f(x), (c) lim f(x), and (d) f(0). x→0+ x→0− x→0 f(x)={15 - x², for x ≤ 0; 15 + x², for x > 0} (a) select the correct choice below and fill in any answer boxes in your choice. a. lim f(x)=□ x→0+ b. the limit does not exist. (b) select the correct choice below and fill in any answer boxes in your choice. a. lim f(x)=□ x→0− b. the limit does not exist. (c) select the correct choice below and fill in any answer boxes in your choice. a. lim f(x)=□ x→0 b. the limit does not exist. (d) select the correct choice below and fill in any answer boxes in your choice. a. f(0)=□ b. the value does not exist.

Answer

Explanation:

Step1: Find right - hand limit

For $x\to0^{+}$, use $f(x)=15 + x^{2}$. Substitute $x = 0$ into $15 + x^{2}$. $\lim_{x\to0^{+}}f(x)=\lim_{x\to0^{+}}(15 + x^{2})=15+0^{2}=15$

Step2: Find left - hand limit

For $x\to0^{-}$, use $f(x)=15 - x^{2}$. Substitute $x = 0$ into $15 - x^{2}$. $\lim_{x\to0^{-}}f(x)=\lim_{x\to0^{-}}(15 - x^{2})=15-0^{2}=15$

Step3: Find overall limit

Since $\lim_{x\to0^{+}}f(x)=\lim_{x\to0^{-}}f(x)=15$, then $\lim_{x\to0}f(x)=15$.

Step4: Find $f(0)$

For $x = 0$, use $f(x)=15 - x^{2}$ (because $x = 0$ satisfies $x\leq0$). Substitute $x = 0$ into $15 - x^{2}$. $f(0)=15-0^{2}=15$

Answer:

(A) A. $\lim_{x\to0^{+}}f(x)=15$ (B) A. $\lim_{x\to0^{-}}f(x)=15$ (C) A. $\lim_{x\to0}f(x)=15$ (D) A. $f(0)=15$