if limx→3 f(x)=3 and limx→3 g(x)=81, find limx→3 ((f(x))^6 √4{g(x)})

Explanation:

Step1: Apply limit - product rule

$\lim_{x\rightarrow3}((f(x))^{6}\sqrt[4]{g(x)})=\lim_{x\rightarrow3}(f(x))^{6}\cdot\lim_{x\rightarrow3}\sqrt[4]{g(x)}$

Step2: Apply limit - power rule for $\lim_{x\rightarrow3}(f(x))^{6}$

$\lim_{x\rightarrow3}(f(x))^{6}=(\lim_{x\rightarrow3}f(x))^{6}$ Since $\lim_{x\rightarrow3}f(x) = 3$, then $(\lim_{x\rightarrow3}f(x))^{6}=3^{6}=729$

Step3: Apply limit - root rule for $\lim_{x\rightarrow3}\sqrt[4]{g(x)}$

$\lim_{x\rightarrow3}\sqrt[4]{g(x)}=\sqrt[4]{\lim_{x\rightarrow3}g(x)}$ Since $\lim_{x\rightarrow3}g(x)=81$, then $\sqrt[4]{\lim_{x\rightarrow3}g(x)}=\sqrt[4]{81}=3$

Step4: Calculate the final limit

$\lim_{x\rightarrow3}(f(x))^{6}\cdot\lim_{x\rightarrow3}\sqrt[4]{g(x)}=729\times3 = 2187$

Answer:

$2187$