let lim f(x)=2. use the limit rules to find the limit given below. x→7 lim 4^f(x) x→7 lim 4^f(x)=□ x→7…
let lim f(x)=2. use the limit rules to find the limit given below. x→7 lim 4^f(x) x→7 lim 4^f(x)=□ x→7 (simplify your answer.)
Answer
Explanation:
Step1: Recall limit - exponent rule
If $\lim_{x\rightarrow a}f(x) = L$ and $y = a^{f(x)}$ where $a>0$, then $\lim_{x\rightarrow a}a^{f(x)}=a^{\lim_{x\rightarrow a}f(x)}$. Here $a = 4$, $f(x)$ is the given function, and $\lim_{x\rightarrow 7}f(x)=2$.
Step2: Substitute the limit value
We have $\lim_{x\rightarrow 7}4^{f(x)}=4^{\lim_{x\rightarrow 7}f(x)}$. Since $\lim_{x\rightarrow 7}f(x) = 2$, then $4^{\lim_{x\rightarrow 7}f(x)}=4^{2}$.
Step3: Calculate the result
$4^{2}=16$.
Answer:
16