Find $\lim _ { x \rightarrow 5 } [ g ( x ) - f ( x ) ]$ for $f ( x ) = 3 x ^ { 3 } \text { and } g ( x ) = \frac { \sqrt { x ^ { 2 } + 6 } } { 6 x ^ { 2 } }$ .
A) $\frac { \sqrt { 31 } } { 150 } - 375$
B) $\frac { \sqrt { 31 } } { 150 } + 375$
C) $\frac { 1 } { 150 } - 375$
D) $\frac { \sqrt { 31 } } { 375 } - 150$
E)limit does not exist

Question

Quizplus · Accepted Answer

To find the limit as $x$ approaches 5 of $g(x) - f(x)$, we substitute $x = 5$ into each function and calculate the difference. For $f(x) = 3x^3$, when $x = 5$, $f(5) = 3(5)^3 = 375$. For $g(x) = \frac{\sqrt{x^2 + 6}}{6x^2}$, when $x = 5$, $g(5) = \frac{\sqrt{5^2 + 6}}{6(5)^2} = \frac{\sqrt{31}}{150}$. Therefore, the limit is $\frac{\sqrt{31}}{150} - 375$.

Find lim⁡x→5[g(x)−f(x)]\lim _ { x \rightarrow 5 } [ g ( x ) - f ( x ) ]limx→5​[g(x)−f(x)]

Find $\lim _ { x \rightarrow 5 } [ g ( x ) - f ( x ) ]$