1. Asymptotes

Video coming soon2 worked examples

Theory

For a rational function, a vertical asymptote occurs where the denominator is zero (and the numerator is not). The non-vertical asymptote describes behaviour as $x \to \pm\infty$ :

Equal degrees on top and bottom → a horizontal asymptote at the ratio of the leading coefficients.
Numerator one degree higher → a slant (oblique) asymptote, found by algebraic division.

The Golden Rule: vertical asymptotes come from the zeros of the denominator; for the non-vertical asymptote, compare the degrees — divide if the top degree is the larger.

⚠️ Common Examiner Traps

Hidden hole: if a factor cancels, there is a hole, not an asymptote — check the numerator is non-zero there.
Slant asymptote: needs polynomial division; the asymptote is the quotient (ignoring the remainder term).
Wrong horizontal value: for equal degrees it is the ratio of the leading coefficients.

Worked examples

Example 1

Find the asymptotes of $y = \dfrac{3x - 2}{x + 1}$ .

Step 1: The denominator is zero at $x = -1$ , giving a vertical asymptote $x = -1$ .

Step 2: The degrees are equal, so as $x \to \pm\infty$ the function tends to the ratio of leading coefficients:

y \to \frac{3x}{x} = 3

So there is a horizontal asymptote $y = 3$ .

Check by division: $\dfrac{3x-2}{x+1} = 3 - \dfrac{5}{x+1}$ , and the fraction vanishes as $x \to \pm\infty$ , confirming $y = 3$ .

Example 2

Find the asymptotes of $y = \dfrac{x^2 + 1}{x - 1}$ .

Step 1: Vertical asymptote where the denominator is zero: $x = 1$ .

Step 2: The numerator's degree is one higher, so divide to find the slant asymptote:

\frac{x^2 + 1}{x - 1} = x + 1 + \frac{2}{x - 1}

Step 3: As $x \to \pm\infty$ the remainder term vanishes, so the slant asymptote is $y = x + 1$ .