Question

question 10 · 1 point
which functions have graphs with no horizontal asymptote and no slant asymptote? select all that apply.
select all that apply:
□ ( f(x) = \frac{x^2}{x - 3} )
□ ( f(x) = \frac{3x}{5x + 4} )
□ ( f(x) = \frac{8x^3}{x + 2} )
□ ( f(x) = \frac{x^4 + 4x}{x^5 - 3} )
□ ( f(x) = \frac{2x^5}{x^3 - 3} )

Explanation:

Step1: Recall asymptote rules for rational functions

For $f(x)=\frac{N(x)}{D(x)}$:

• Horizontal asymptote (HA): exists if $\deg(N)\leq\deg(D)$
• Slant asymptote (SA): exists if $\deg(N)=\deg(D)+1$

Step2: Analyze each function

1. $f(x)=\frac{x^2}{x-3}$: $\deg(N)=2$, $\deg(D)=1$ → $\deg(N)=\deg(D)+1$ → SA exists.
2. $f(x)=\frac{3x}{5x+4}$: $\deg(N)=1$, $\deg(D)=1$ → HA exists ($y=\frac{3}{5}$).
3. $f(x)=\frac{8x^3}{x+2}$: $\deg(N)=3$, $\deg(D)=1$ → $\deg(N)>\deg(D)+1$ → no HA, no SA.
4. $f(x)=\frac{x^4+4x}{x^5-3}$: $\deg(N)=4$, $\deg(D)=5$ → HA exists ($y=0$).
5. $f(x)=\frac{2x^5}{x^3-3}$: $\deg(N)=5$, $\deg(D)=3$ → $\deg(N)>\deg(D)+1$ → no HA, no SA.

Answer:

$f(x)=\frac{8x^3}{x+2}$, $f(x)=\frac{2x^5}{x^3-3}$