Determine the horizontal asymptote of the graph of the function. $$ f(x)=\frac{3 x^{3}+5}{2 x^{3}-3} $$

Question

UpStudy AI · Accepted Answer

1. Identify the highest degree in the numerator and denominator. The highest degree is 3 in both $$3x^3$$ (numerator) and $$2x^3$$ (denominator).

2. Divide both the numerator and the denominator by $$x^3$$:

$$
   f(x)=\frac{3+\frac{5}{x^3}}{2-\frac{3}{x^3}}
   $$

3. As $$ x $$ approaches infinity, the terms $$\frac{5}{x^3}$$ and $$\frac{3}{x^3}$$ approach 0:

$$
   \lim_{x \to \infty} f(x)=\frac{3+0}{2-0}=\frac{3}{2}
   $$

4. The horizontal asymptote of the function is therefore:

$$
   y=\frac{3}{2}
   $$
The horizontal asymptote of the function $$ f(x)=\frac{3x^{3}+5}{2x^{3}-3} $$ is $$ y=\frac{3}{2} $$.

Determine the horizontal asymptote of the graph of the function. \( f(x)=\frac{3 x^{3}+5}{2 x^{3}-3} \)