the graph of $f(x)=\frac{-5x^{4}-3x + 1}{x^{5}+x^{2}-5}$ will behave like which function for large values of…

Explanation:

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

For large values of (|x|), divide both the numerator and denominator of (f(x)=\frac{-5x^{4}-3x + 1}{x^{5}+x^{2}-5}) by (x^{5}). We get (f(x)=\frac{\frac{-5x^{4}}{x^{5}}-\frac{3x}{x^{5}}+\frac{1}{x^{5}}}{\frac{x^{5}}{x^{5}}+\frac{x^{2}}{x^{5}}-\frac{5}{x^{5}}}=\frac{-\frac{5}{x}-\frac{3}{x^{4}}+\frac{1}{x^{5}}}{1 + \frac{1}{x^{3}}-\frac{5}{x^{5}}}).

Step2: Evaluate the limit as (|x|\to\infty)

As (|x|\to\infty), (\lim_{|x|\to\infty}\frac{-\frac{5}{x}-\frac{3}{x^{4}}+\frac{1}{x^{5}}}{1 + \frac{1}{x^{3}}-\frac{5}{x^{5}}}). Since (\lim_{|x|\to\infty}\frac{a}{x^{n}} = 0) for (a) constant and (n>0), we have (\lim_{|x|\to\infty}\frac{-\frac{5}{x}-\frac{3}{x^{4}}+\frac{1}{x^{5}}}{1 + \frac{1}{x^{3}}-\frac{5}{x^{5}}}=\frac{0 - 0+0}{1 + 0 - 0}=0).

Answer:

(y = 0)