Limits and Continuity
Grade 12 · Math · Study Guide
Master the foundational concept of limits and understand what it means for a function to be continuous.
A limit describes the value a function approaches as the input gets closer and closer to some number. We write lim(x→a) f(x) = L to mean f(x) gets arbitrarily close to L as x approaches a, even if f(a) is undefined.
A function f is continuous at x = a if three conditions hold: f(a) exists, lim(x→a) f(x) exists, and those two values are equal. If any condition fails, the function has a discontinuity at that point.
Common discontinuities include removable holes (the limit exists but f(a) doesn't match), jump discontinuities (left and right limits differ), and infinite discontinuities (the function blows up to ±∞).
| Law | Statement |
|---|---|
| Sum | lim(f + g) = lim f + lim g |
| Product | lim(f · g) = lim f · lim g |
| Quotient | lim(f / g) = lim f / lim g, provided lim g ≠ 0 |
| Constant | lim(c · f) = c · lim f |
| Type | Characteristic |
|---|---|
| Removable | Limit exists; point missing or mismatched |
| Jump | Left and right limits exist but are unequal |
| Infinite | Function grows without bound near the point |
Key terms
- Limit
- The value a function approaches as x nears a given point.
- Continuity
- A function is continuous at a point when its limit equals its value there.
- One-sided limit
- Limit taken from only the left (x→a⁻) or only the right (x→a⁺).
- Removable discontinuity
- A hole in the graph where the limit exists but the function value differs or is missing.
Practice questions
What is lim(x→2) (x² - 4)/(x - 2)?
- 0
- 2
- 4
- Undefined
Show answer
Factor: (x²-4)/(x-2) = (x+2)(x-2)/(x-2) = x+2. Substituting x=2 gives 4.
Which condition is NOT required for f to be continuous at x = a?
- f(a) is defined
- lim(x→a) f(x) exists
- f is differentiable at x = a
- lim(x→a) f(x) = f(a)
Show answer
Differentiability is a stronger condition than continuity and is not required for continuity.
Evaluate lim(x→∞) (3x² + 1)/(x² - 5).
Show answer
Divide numerator and denominator by x²: (3 + 1/x²)/(1 - 5/x²) → 3/1 = 3 as x→∞.
If lim(x→a) f(x) = 5 but f(a) = 7, then f has a removable discontinuity at x = a.
Show answer
The limit exists (5) but doesn't equal f(a) (7), so there is a removable discontinuity.
What is lim(x→0⁺) 1/x?
Show answer
As x approaches 0 from the right, 1/x grows without bound toward positive infinity.
Fun facts
- The formal epsilon-delta definition of a limit was developed by Cauchy and Weierstrass in the 19th century, finally giving calculus a rigorous foundation.
- Limits let mathematicians handle division by zero indirectly — lim(x→0) sin(x)/x = 1, even though sin(0)/0 is undefined.
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