Question

due wednesday by 11:59pm points 100 submitting an external tool which functions have graphs with a slant asymptote? select all that apply. select all that apply: $f(x)=\frac{x + 1}{x^{2}}$ $f(x)=\frac{3x^{2}}{5x}$ $f(x)=\frac{x^{4}}{2x^{2}}$ $f(x)=\frac{2x^{4}+1}{x^{3}}$ $f(x)=\frac{3x^{3}}{x^{4}+1}$

Explanation:

Step1: Recall slant - asymptote condition

A rational function $y = \frac{f(x)}{g(x)}$ has a slant asymptote when the degree of the numerator $f(x)$ is exactly one more than the degree of the denominator $g(x)$.

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

Degree of numerator is 1, degree of denominator is 2. Since $1<2$, no slant asymptote.

Step3: Analyze $f(x)=\frac{3x^{2}}{5x}$

Degree of numerator is 2, degree of denominator is 1. Simplify $f(x)=\frac{3}{5}x$, which is a linear function (not an asymptote in the non - trivial sense for rational functions, but it shows the behavior). It has a slant asymptote.

Step4: Analyze $f(x)=\frac{x^{4}}{2x^{2}}$

Degree of numerator is 4, degree of denominator is 2. Since $4-2 = 2>1$, no slant asymptote.

Step5: Analyze $f(x)=\frac{2x^{4}+1}{x^{3}}$

Degree of numerator is 4, degree of denominator is 3. Since $4 - 3=1$, it has a slant asymptote.

Step6: Analyze $f(x)=\frac{3x^{3}}{x^{4}+1}$

Degree of numerator is 3, degree of denominator is 4. Since $3<4$, no slant asymptote.

Answer:

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