Answer
$$\lim _{x\rightarrow \infty }\dfrac {3x^{3}-7}{x^{4}+5x^{2}}=0;\lim _{x\rightarrow -\infty }\dfrac {3x^{3}-7}{x^{4}+5x^{2}}=0;y=0$$
Work Step by Step
$$\lim _{x\rightarrow \infty }\dfrac {3x^{3}-7}{x^{4}+5x^{2}}=\dfrac {\dfrac {3}{x}-\dfrac {7}{x^{4}}}{1+\dfrac {5}{x^{2}}}=\dfrac {0-0}{1+0}=\dfrac {0}{1}=0$$
$$\lim _{x\rightarrow -\infty }\dfrac {3x^{3}-7}{x^{4}+5x^{2}}=\dfrac {\dfrac {3}{x}-\dfrac {7}{x^{4}}}{1+\dfrac {5}{x^{2}}}=\dfrac {0-0}{1+0}=\dfrac {0}{1}=0$$
So the horizontal asymptote will be $y=0$
