Where the denominator vanishes: A fraction explodes when its bottom is zero. So a rational function has a vertical asymptote wherever the denominator = 0 (and the top isn't also 0 there).
Set the denominator to 0 and solve.
IB-style question — vertical asymptotes
Find the vertical asymptotes of f(x) = (2x + 1)/(x² − x − 6).
Step by step
- Factorise and set the denominator to 0.
- Solve.
Final answer
Vertical asymptotes x = 3 and x = −2.
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What happens for large x: For very large x, only the highest powers matter:
top degree < bottom degree → y = 0.
equal degrees → y = (ratio of leading coefficients).
IB-style question — horizontal asymptotes
Find the horizontal asymptote of (a) f(x) = (3x + 1)/(x² + 1) and (b) g(x) = (2x² + 1)/(x² − 4).
Step by step
- (a) Top degree 1 < bottom degree 2, so y → 0.
- (b) Equal degrees: divide the leading coefficients 2 ÷ 1.
Final answer
(a) y = 0; (b) y = 2.