Find
$$\lim _ { x \rightarrow 3 } [ g ( x ) - f ( x ) ]$$
for $f ( x ) = 3 x ^ { 3 }$ and $g ( x ) = \frac { \sqrt { x ^ { 2 } + 2 } } { 6 x ^ { 2 } }$.
A) $\frac { \sqrt { 11 } } { 54 } - 81$
B) $\frac { \sqrt { 3 } } { 6 } - 81$
C) $\frac { 3 \sqrt { 11 } } { 2 }$
D) $\frac { \sqrt { 3 } } { 6 } + 81$
E) limit does not exist

Question

Quizplus · Accepted Answer

To find the limit as $x$ approaches 3 of $g(x) - f(x)$, we substitute $x = 3$ into each function and calculate the difference. For $f(x) = 3x^3$, when $x = 3$, $f(3) = 3(3)^3 = 3 \cdot 27 = 81$. For $g(x) = \frac{\sqrt{x^2 + 2}}{6x^2}$, when $x = 3$, $g(3) = \frac{\sqrt{3^2 + 2}}{6 \cdot 3^2} = \frac{\sqrt{9 + 2}}{54} = \frac{\sqrt{11}}{54}$. Therefore, the limit is $\frac{\sqrt{11}}{54} - 81$.

Find lim⁡x→3[g(x)−f(x)]\lim _ { x \rightarrow 3 } [ g ( x ) - f ( x ) ]x→3lim​[g(x)−f(x)]

Find $\lim _ { x \rightarrow 3 } [ g ( x ) - f ( x ) ]$