Question

find each function value and limit. use -∞ or ∞ where appropriate.
f(x) = \\(\frac{7x^{4}-14x^{2}}{12x^{5}+6}\\)
(a) f(-6)
(b) f(-12)
(c) \\(\lim_{x\to -\infty} f(x)\\)
(a) f(-6)=\\(\square\\)
(round to the nearest thousandth as needed.)
(b) f(-12)=\\(\square\\)
(round to the nearest thousandth as needed.)
(c) select the correct choice below and, if necessary, fill in the answer box to complete your choice.
○a. \\(\lim_{x\to -\infty}\frac{7x^{4}-14x^{2}}{12x^{5}+6}=\\(\square\\)
(type an integer or a decimal.)
○b. the limit does not exist.

Explanation:

Step1: Calculate f(-6)

Substitute \(x = - 6\) into \(f(x)=\frac{7x^{4}-14x^{2}}{12x^{5}+6}\).
\[

$$\begin{align*} f(-6)&=\frac{7\times(-6)^{4}-14\times(-6)^{2}}{12\times(-6)^{5}+6}\\ &=\frac{7\times1296 - 14\times36}{12\times(-7776)+6}\\ &=\frac{9072-504}{-93312 + 6}\\ &=\frac{8568}{-93306}\\ &\approx - 0.092 \end{align*}$$

\]

Step2: Calculate f(-12)

Substitute \(x=-12\) into \(f(x)=\frac{7x^{4}-14x^{2}}{12x^{5}+6}\).
\[

$$\begin{align*} f(-12)&=\frac{7\times(-12)^{4}-14\times(-12)^{2}}{12\times(-12)^{5}+6}\\ &=\frac{7\times20736-14\times144}{12\times(-248832)+6}\\ &=\frac{145152 - 2016}{-2985984+6}\\ &=\frac{143136}{-2985978}\\ &\approx - 0.048 \end{align*}$$

\]

Step3: Calculate \(\lim_{x

ightarrow-\infty}f(x)\)
For the rational - function \(f(x)=\frac{7x^{4}-14x^{2}}{12x^{5}+6}\), when \(x
ightarrow-\infty\), the degree of the numerator \(n = 4\) and the degree of the denominator \(m = 5\), and \(n\lt m\).
So \(\lim_{x
ightarrow-\infty}\frac{7x^{4}-14x^{2}}{12x^{5}+6}=0\)

Answer:

(A) \(-0.092\)
(B) \(-0.048\)
(C) A. \(0\)