find each function value and limit. use -∞ or ∞ where appropriate.\nf(x) = \\(\\frac{7x^{4}-14x^{2}}{12x^{5}+…

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

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

Answer

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\rightarrow-\infty}f(x))

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

Answer:

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