find each function value and limit. use -∞ or ∞ where appropriate.\nf(x)=\frac{5x^{4}-10x^{2}}{10x^{5}+5}\n(a…

find each function value and limit. use -∞ or ∞ where appropriate.\nf(x)=\frac{5x^{4}-10x^{2}}{10x^{5}+5}\n(a) f(-6)\n(b) f(-12)\n(c) lim_{x\rightarrow - infty} f(x)\n(a) f(-6)= \n(round to the nearest thousandth as needed.)\n(b) f(-12)= \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\rightarrow - infty}\frac{5x^{4}-10x^{2}}{10x^{5}+5}= \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{5x^{4}-10x^{2}}{10x^{5}+5}\n(a) f(-6)\n(b) f(-12)\n(c) lim_{x\rightarrow - infty} f(x)\n(a) f(-6)= \n(round to the nearest thousandth as needed.)\n(b) f(-12)= \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\rightarrow - infty}\frac{5x^{4}-10x^{2}}{10x^{5}+5}= \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{5x^{4}-10x^{2}}{10x^{5}+5}). [ \begin{align*} f(-6)&=\frac{5\times(-6)^{4}-10\times(-6)^{2}}{10\times(-6)^{5}+5}\ &=\frac{5\times1296 - 10\times36}{10\times(-7776)+5}\ &=\frac{6480-360}{-77760 + 5}\ &=\frac{6120}{-77755}\ &\approx - 0.079 \end{align*} ]

Step2: Calculate (f(-12))

Substitute (x=-12) into (f(x)=\frac{5x^{4}-10x^{2}}{10x^{5}+5}). [ \begin{align*} f(-12)&=\frac{5\times(-12)^{4}-10\times(-12)^{2}}{10\times(-12)^{5}+5}\ &=\frac{5\times20736-10\times144}{10\times(-248832)+5}\ &=\frac{103680 - 1440}{-2488320+5}\ &=\frac{102240}{-2488315}\ &\approx - 0.041 \end{align*} ]

Step3: Find (\lim_{x\rightarrow-\infty}f(x))

Divide both the numerator and denominator by (x^{5}): [ \begin{align*} \lim_{x\rightarrow-\infty}\frac{5x^{4}-10x^{2}}{10x^{5}+5}&=\lim_{x\rightarrow-\infty}\frac{\frac{5x^{4}}{x^{5}}-\frac{10x^{2}}{x^{5}}}{\frac{10x^{5}}{x^{5}}+\frac{5}{x^{5}}}\ &=\lim_{x\rightarrow-\infty}\frac{\frac{5}{x}-\frac{10}{x^{3}}}{10+\frac{5}{x^{5}}} \end{align*} ] As (x\rightarrow-\infty), (\frac{5}{x}\rightarrow0), (\frac{10}{x^{3}}\rightarrow0) and (\frac{5}{x^{5}}\rightarrow0). So (\lim_{x\rightarrow-\infty}\frac{\frac{5}{x}-\frac{10}{x^{3}}}{10+\frac{5}{x^{5}}}=0)

Answer:

(A) (f(-6)\approx - 0.079) (B) (f(-12)\approx - 0.041) (C) A. (\lim_{x\rightarrow-\infty}\frac{5x^{4}-10x^{2}}{10x^{5}+5}=0)