Find $\lim _ { x \rightarrow + \infty } f ( x )$ and $\lim _ { x \rightarrow - \infty } f ( x )$ . If the limiting value is infinite indicate whether it is +$\infty$or -$\infty$. $f ( x ) = \frac { 1 - 4 x ^ { 3 } } { 5 x ^ { 3 } - 9 x - 3 }$
A) $- \frac { 4 } { 5 } , \frac { 4 } { 5 }$
B) $- \frac { 4 } { 5 } , - \frac { 4 } { 5 }$
C) +$\infty$,-$\infty$
D) -$\infty$,+$\infty$

Question

Quizplus · Accepted Answer

To find the limits as $x$ approaches $+\infty$ and $-\infty$, we look at the leading terms in the numerator and denominator since they dominate the behavior of the function at large magnitudes of $x$. The leading term in the numerator is $-4x^3$ and in the denominator is $5x^3$. Thus, for both limits, the function simplifies to $\frac{-4x^3}{5x^3} = -\frac{4}{5}$, indicating that as $x$ approaches both $+\infty$ and $-\infty$, the function approaches $-\frac{4}{5}$.

Find lim⁡x→+∞f(x)\lim _ { x \rightarrow + \infty } f ( x )limx→+∞​f(x)

Find $\lim _ { x \rightarrow + \infty } f ( x )$