Find $\lim _ { x \rightarrow 5 } [ g ( x ) - f ( x ) ]$ for $f ( x ) = 5 x ^ { 3 }$ and $g ( x ) = \frac { \sqrt { x ^ { 2 } + 9 } } { 3 x ^ { 2 } }$ .
A) $\frac { \sqrt { 34 } } { 75 } - 625$
B) $\frac { \sqrt { 34 } } { 75 } + 625$
C) $\frac { \sqrt { 34 } } { 75 }$
D)limit does not exist
E) $\frac { 1 } { 75 } - 625$

Question

Quizplus · Accepted Answer

To find the limit as $x$ approaches 5 of $g(x) - f(x)$, substitute $x = 5$ into each function and perform the subtraction. For $f(x) = 5x^3$, when $x = 5$, $f(5) = 5(5)^3 = 5(125) = 625$. For $g(x) = \frac{\sqrt{x^2 + 9}}{3x^2}$, when $x = 5$, $g(5) = \frac{\sqrt{5^2 + 9}}{3(5)^2} = \frac{\sqrt{25 + 9}}{75} = \frac{\sqrt{34}}{75}$. Therefore, the limit is $\frac{\sqrt{34}}{75} - 625$.

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

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