determine lim f(x) and lim f(x) for the following function. then give the horizontal asymptotes of f, if…

determine lim f(x) and lim f(x) for the following function. then give the horizontal asymptotes of f, if any.\n f(x)=\frac{5x^{3}-7}{x^{4}+3x^{2}} \nevaluate lim f(x) select the correct choice below and, if necessary, fill in the answer box to complete your choice\n lim_{x \to infty} \noa\n lim_{x \to infty}\frac{5x^{3}-7}{x^{4}+3x^{2}}=square \text{ (simplify your answer.)} \nob. the limit does not exist and is neither (infty) nor (-infty).

determine lim f(x) and lim f(x) for the following function. then give the horizontal asymptotes of f, if any.\n f(x)=\frac{5x^{3}-7}{x^{4}+3x^{2}} \nevaluate lim f(x) select the correct choice below and, if necessary, fill in the answer box to complete your choice\n lim_{x \to infty} \noa\n lim_{x \to infty}\frac{5x^{3}-7}{x^{4}+3x^{2}}=square \text{ (simplify your answer.)} \nob. the limit does not exist and is neither (infty) nor (-infty).

Answer

Explanation:

Step1: Divide numerator and denominator by highest - power of x

Divide both the numerator and denominator of $f(x)=\frac{5x^{3}-7}{x^{4}+3x^{2}}$ by $x^{4}$. We get $\lim_{x\rightarrow\infty}\frac{\frac{5x^{3}}{x^{4}}-\frac{7}{x^{4}}}{\frac{x^{4}}{x^{4}}+\frac{3x^{2}}{x^{4}}}=\lim_{x\rightarrow\infty}\frac{\frac{5}{x}-\frac{7}{x^{4}}}{1 + \frac{3}{x^{2}}}$.

Step2: Evaluate the limit of each term

As $x\rightarrow\infty$, we know that $\lim_{x\rightarrow\infty}\frac{5}{x}=0$, $\lim_{x\rightarrow\infty}\frac{7}{x^{4}} = 0$, and $\lim_{x\rightarrow\infty}\frac{3}{x^{2}}=0$. So, $\lim_{x\rightarrow\infty}\frac{\frac{5}{x}-\frac{7}{x^{4}}}{1+\frac{3}{x^{2}}}=\frac{0 - 0}{1+0}=0$.

Answer:

A. $\lim_{x\rightarrow\infty}\frac{5x^{3}-7}{x^{4}+3x^{2}}=0$