Rational Function Asymptotes

Finding asymptotes of $f(x) = \dfrac{2x + 1}{x - 3}$

Vertical asymptote (VA)

Rule: Set the denominator equal to zero (after canceling common factors).

$x - 3 = 0 \implies x = 3$

VA: $x = 3$. The function blows up to $\pm\infty$ as $x$ approaches 3.

Horizontal asymptote (HA)

Rule: Compare the degree of the numerator vs denominator:

• Same degree → HA is the ratio of leading coefficients.
• Numerator degree < denominator → HA is $y = 0$.
• Numerator degree > denominator → no horizontal asymptote (oblique instead).

Here: numerator degree = 1, denominator degree = 1 (same). Leading coefficients: $2/1 = 2$.

HA: $y = 2$. As $x \to \pm\infty$, $f(x) \to 2$.

Intercepts

x-intercept: Set numerator = 0: $2x + 1 = 0 \implies x = -1/2$. Point: $(-1/2, 0)$.

y-intercept: $f(0) = \frac{1}{-3} = -1/3$. Point: $(0, -1/3)$.

The graph behavior

The curve approaches but never touches the dashed asymptote lines. Near $x = 3$, the function shoots to $+\infty$ on one side and $-\infty$ on the other. Far from the origin, the curve flattens toward $y = 2$.

Try it in the visualization

Adjust the numerator and denominator coefficients. The asymptotes update automatically. The curve shows the approaching behavior on each side.