for the given piece - wise function, use properties of limits to find the indicated limit, or state that the…
for the given piece - wise 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) f(x)={x + 6 if x < 3; x + 7 if x ≥ 3} 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)= (simplify your answer.) b. lim f(x) does not exist.
Answer
Explanation:
Step1: Analyze left - hand limit
For $\lim_{x\rightarrow3^{-}}f(x)$, when $x\rightarrow3^{-}$, we use the part of the piece - wise function where $x < 3$. So $f(x)=x + 6$. Substitute $x = 3$ into $x+6$, we get $\lim_{x\rightarrow3^{-}}f(x)=3 + 6=9$.
Step2: Analyze right - hand limit
For $\lim_{x\rightarrow3^{+}}f(x)$, when $x\rightarrow3^{+}$, we use the part of the piece - wise function where $x\geq3$. So $f(x)=x + 7$. Substitute $x = 3$ into $x + 7$, we get $\lim_{x\rightarrow3^{+}}f(x)=3+7 = 10$.
Answer:
a. A. $\lim_{x\rightarrow3^{-}}f(x)=9$ b. A. $\lim_{x\rightarrow3^{+}}f(x)=10$