Answer
If the graph of a function f approaches b as x increases or decreases without bound, then the line $y=b$ is a horizontal asymptote of the graph of f.
The equation of such a line for the graph of $f\left( x \right)=\frac{x-10}{3{{x}^{2}}+x+1}$ is $y=0$ the equation of such a line for the graph of $y=\frac{{{x}^{2}}-10}{3{{x}^{2}}+x+1}$ is $y=\frac{1}{3}.$
Work Step by Step
The line $y=b$ is a horizontal asymptote of the graph of a function f if $f\left( x \right)$ approaches b as x increases or decreases without bound.
For the function $f\left( x \right)=\frac{x-10}{3{{x}^{2}}+x+1}$ the degree of the numerator is 1 which is less than the degree of the denominator (2).
Thus, the graph of f has the x-axis as a horizontal asymptote.
The equation of the horizontal asymptote is $y=0$.
For function $y=\frac{{{x}^{2}}-10}{3{{x}^{2}}+x+1}$, the degree of the numerator is 2, which is equal to the degree of the denominator.
The leading coefficients of the numerator and denominator, 1 and 3 respectively, are used to obtain the equation of the horizontal asymptote.
The horizontal asymptote is $y=\frac{1}{3}$.
