consider f(x) = (-14x^4 + 2x + 18)/(x^4 - 3x^2). which statement represents the correct interpretation of…

consider f(x) = (-14x^4 + 2x + 18)/(x^4 - 3x^2). which statement represents the correct interpretation of this limit? lim(x→±∞) f(x) the limit shows that as x→±∞, f(x)→14. the limit shows that as x→±∞, f(x)→ - 14. the limit shows that as x→±∞, f(x)→ - ∞. the limit shows that as x→±∞, f(x)→0.

consider f(x) = (-14x^4 + 2x + 18)/(x^4 - 3x^2). which statement represents the correct interpretation of this limit? lim(x→±∞) f(x) the limit shows that as x→±∞, f(x)→14. the limit shows that as x→±∞, f(x)→ - 14. the limit shows that as x→±∞, f(x)→ - ∞. the limit shows that as x→±∞, f(x)→0.

Answer

Explanation:

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

For (f(x)=\frac{- 14x^{4}+2x + 18}{x^{4}-3x^{2}}), divide each term by (x^{4}). We get (\lim_{x\rightarrow\pm\infty}\frac{-14+\frac{2}{x^{3}}+\frac{18}{x^{4}}}{1 - \frac{3}{x^{2}}}).

Step2: Evaluate the limit of each term.

As (x\rightarrow\pm\infty), (\lim_{x\rightarrow\pm\infty}\frac{2}{x^{3}} = 0), (\lim_{x\rightarrow\pm\infty}\frac{18}{x^{4}}=0) and (\lim_{x\rightarrow\pm\infty}\frac{3}{x^{2}} = 0).

Step3: Calculate the overall limit.

(\lim_{x\rightarrow\pm\infty}\frac{-14+\frac{2}{x^{3}}+\frac{18}{x^{4}}}{1 - \frac{3}{x^{2}}}=\frac{-14 + 0+0}{1-0}=-14).

Answer:

The limit shows that as (x\rightarrow\pm\infty), (f(x)\rightarrow - 14).