Question

question
which functions have graphs with a horizontal asymptote? select all that apply.
select all that apply:
$f(x)=\frac{x^{3}+1}{x^{2}}$
$f(x)=\frac{2x - 1}{x^{2}+1}$
$f(x)=\frac{6x + 4}{2x - 1}$
$f(x)=\frac{2x^{3}}{x^{5}}$
$f(x)=\frac{4x^{4}}{2x^{2}-3}$

Explanation:

Step1: Recall horizontal - asymptote rules for rational functions

For a rational function $f(x)=\frac{a_nx^n + a_{n - 1}x^{n-1}+\cdots+a_0}{b_mx^m + b_{m - 1}x^{m-1}+\cdots+b_0}$, if $n\lt m$, the horizontal asymptote is $y = 0$; if $n=m$, the horizontal asymptote is $y=\frac{a_n}{b_m}$; if $n>m$, there is no horizontal asymptote.

Step2: Analyze $f(x)=\frac{x^3 + 1}{x^2}$

Here $n = 3$ and $m = 2$ ($n>m$), so there is no horizontal asymptote.

Step3: Analyze $f(x)=\frac{2x-1}{x^2 + 1}$

Here $n = 1$ and $m = 2$ ($n

Step4: Analyze $f(x)=\frac{6x + 4}{2x-1}$

Here $n = 1$ and $m = 1$, and $\frac{a_n}{b_m}=\frac{6}{2}=3$, so the horizontal asymptote is $y = 3$.

Step5: Analyze $f(x)=\frac{2x^3}{x^5}$

Here $n = 3$ and $m = 5$ ($n

Step6: Analyze $f(x)=\frac{4x^4}{2x^2-3}$

Here $n = 4$ and $m = 2$ ($n>m$), so there is no horizontal asymptote.

Answer:

$f(x)=\frac{2x - 1}{x^2+1}$, $f(x)=\frac{6x + 4}{2x-1}$, $f(x)=\frac{2x^3}{x^5}$