Lesson 3 of 18Article16 min
Continuity, Asymptotes & Infinite Limits
f is continuous at a if limₓ→ₐ f(x) exists, f(a) is defined, and they are equal. Graphically: you can draw through a without lifting the pen.
Continuity checklist
f is continuous at a if limₓ→ₐ f(x) exists, f(a) is defined, and they are equal. Graphically: you can draw through a without lifting the pen.
- Removable discontinuity: limit exists but f(a) missing or wrong — fill the hole.
- Jump discontinuity: left and right limits differ.
- Infinite discontinuity: function blows up near a vertical asymptote.
End behaviour
As x → ∞, dominant terms decide the limit. For rational functions, compare degrees of numerator and denominator.
- deg(num) < deg(den) → horizontal asymptote y = 0.
- deg(num) = deg(den) → y = ratio of leading coefficients.
- deg(num) > deg(den) → no horizontal asymptote (may have oblique).