Find $\lim _ { x \rightarrow - \infty } \left[ \frac { x } { ( 7 + x ) ^ { 2 } } - 6 \right]$ (if it exists).
A) $- \frac { 5 } { 49 }$
B) 6
C) $- 6$
D) $\infty$
E) limit does not exist

Question

Quizplus · Accepted Answer

To find the limit as $x$ approaches $-\infty$, we can analyze the behavior of each term in the expression. The term $\frac{x}{(7+x)^2}$ approaches 0 as $x$ approaches $-\infty$, because the numerator grows linearly while the denominator grows quadratically. Therefore, the entire expression approaches $-6$, as the $\frac{x}{(7+x)^2}$ term becomes negligible.

Find lim⁡x→−∞[x(7+x)2−6]\lim _ { x \rightarrow - \infty } \left[ \frac { x } { ( 7 + x ) ^ { 2 } } - 6 \right]limx→−∞​[(7+x)2x​−6]

Find $\lim _ { x \rightarrow - \infty } \left[ \frac { x } { ( 7 + x ) ^ { 2 } } - 6 \right]$