for the given piecewise - function, use properties of limits to find the indicated limit, or state that the…
for the given piecewise - function, use properties of limits to find the indicated limit, or state that the limit does not exist. a. lim f(x) b. lim f(x) c. lim f(x) a. find lim f(x). select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. lim f(x)=9 (simplify your answer.) b. lim f(x) does not exist. b. find lim f(x). select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. lim f(x)=10 (simplify your answer.) b. lim f(x) does not exist. c. find lim f(x). select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. lim f(x)= (simplify your answer.) b. lim f(x) does not exist.
Answer
Explanation:
Step1: Analyze left - hand limit as $x\to3^{-}$
For $x < 3$, $f(x)=x + 6$. Substitute $x = 3$ into $x+6$, we get $\lim_{x\to3^{-}}f(x)=3 + 6=9$.
Step2: Analyze right - hand limit as $x\to3^{+}$
For $x>3$, $f(x)=x + 7$. Substitute $x = 3$ into $x + 7$, we get $\lim_{x\to3^{+}}f(x)=3+7 = 10$.
Step3: Analyze two - sided limit as $x\to3$
Since $\lim_{x\to3^{-}}f(x)=9$ and $\lim_{x\to3^{+}}f(x)=10$, and $\lim_{x\to3^{-}}f(x)\neq\lim_{x\to3^{+}}f(x)$, then $\lim_{x\to3}f(x)$ does not exist.
Answer:
a. $\lim_{x\to3^{-}}f(x)=9$ b. $\lim_{x\to3^{+}}f(x)=10$ c. B. $\lim_{x\to3}f(x)$ does not exist