Limits and Continuity Explained: Understanding Behavior Near and At Points
PUC Mathematics | Limits & Continuity | Daniel Classes
Why This Chapter Bridges Algebra and Calculus
In algebra, you ask: what is f(x) at this point? In calculus, you ask: what does f(x) approach as x gets close to this point?
That difference — between “what is” and “what does it approach” — is where limits live. And continuity is the special case where what the function approaches equals what it actually is.
Core insight: A limit describes behavior near a point, not at the point. f(x) can approach a value even if f isn’t defined there.
The Limit Concept
Definition: “The limit of f(x) as x approaches a is L” (written lim(x→a) f(x) = L) means: as x gets arbitrarily close to a (but not equal to a), f(x) gets arbitrarily close to L.
Key phrase: “close to a, but not equal to a.” The limit doesn’t care what happens exactly at a.
Example: f(x) = (x² – 1)/(x – 1)
At x = 1, the function is undefined (0/0). But as x approaches 1:
lim(x→1) (x² – 1)/(x – 1) = lim(x→1) (x + 1) = 2
Even though f(1) doesn’t exist, the limit exists.
Left and Right Limits
Left limit: lim(x→a⁻) f(x) = L means x approaches a from the left (x < a).
Right limit: lim(x→a⁺) f(x) = L means x approaches a from the right (x > a).
Two-sided limit: lim(x→a) f(x) = L exists if and only if lim(x→a⁻) f(x) = lim(x→a⁺) f(x) = L.
Example: f(x) = |x|/x
lim(x→0⁻) |x|/x = lim(x→0⁻) -x/x = -1
lim(x→0⁺) |x|/x = lim(x→0⁺) x/x = 1
Since left ≠ right, lim(x→0) |x|/x does NOT exist.
Continuity
Definition: A function f is continuous at x = a if:
- f(a) is defined,
- lim(x→a) f(x) exists, and
- lim(x→a) f(x) = f(a)
Intuitive meaning: The graph has no jumps, holes, or gaps at x = a.
Example (Continuous): f(x) = x² is continuous everywhere. At any point a: f(a) = a², and lim(x→a) x² = a² = f(a). ✓
Example (Discontinuous): f(x) = |x|/x is discontinuous at x = 0 because lim(x→0) |x|/x doesn’t exist.
Limit Laws
If lim(x→a) f(x) = L and lim(x→a) g(x) = M, then:
Sum: lim(x→a) [f(x) + g(x)] = L + M
Product: lim(x→a) [f(x) × g(x)] = L × M
Quotient: lim(x→a) [f(x)/g(x)] = L/M (if M ≠ 0)
Power: lim(x→a) [f(x)]^n = L^n
Why they matter: They let you evaluate complicated limits by breaking them into simple pieces.
Squeeze Theorem
Statement: If g(x) ≤ f(x) ≤ h(x) near a, and lim(x→a) g(x) = lim(x→a) h(x) = L, then lim(x→a) f(x) = L.
Example: Prove lim(x→0) x sin(1/x) = 0
We know -1 ≤ sin(1/x) ≤ 1, so -|x| ≤ x sin(1/x) ≤ |x|
Since lim(x→0) (-|x|) = lim(x→0) |x| = 0, by the Squeeze Theorem, lim(x→0) x sin(1/x) = 0.
How PUC Exams Test Limits & Continuity
Type 1: Evaluate Limits (2-3 marks)
Example: “Find lim(x→2) (x² + 3x).”
Solution: = 2² + 3(2) = 10 (direct substitution works for continuous functions)
Type 2: Handle Indeterminate Forms (3-5 marks)
Example: “Find lim(x→1) (x² – 1)/(x – 1).”
Solution: Factor: lim(x→1) (x+1)(x-1)/(x-1) = lim(x→1) (x+1) = 2
Type 3: Check Continuity (3-5 marks)
Example: “Is f(x) = {x² if x ≤ 1; 2x if x > 1} continuous at x = 1?”
Solution: f(1) = 1, lim(x→1⁻) x² = 1, lim(x→1⁺) 2x = 2. Since left ≠ right, discontinuous.
The Five Common Mistakes
Mistake 1: Assuming Limits Always Equal Function Values
Example: “Find lim(x→1) (x² – 1)/(x – 1).”
Wrong: “Just plug in x = 1: (1 – 1)/(1 – 1) = 0/0, so there’s no limit.”
Right: “0/0 is indeterminate. Factor: (x+1)(x-1)/(x-1) = x+1, so lim = 2.”
Fix: Limits describe behavior near a point, not at it. If you get 0/0 or ∞/∞, simplify first.
Mistake 2: Confusing Left and Right Limits
Example: “Find lim(x→0⁻) |x|/x.”
Wrong: “It’s 1.” [Using the right-side formula]
Right: “From the left, x < 0, so |x| = -x. Thus (-x)/x = -1."
Fix: For x approaching from the left (x < a), use the left side of the function definition.
Mistake 3: Not Checking All Three Continuity Conditions
Example: “Is f continuous at x = 2?”
Wrong: “If the limit exists, then f is continuous.” [Skipping the check that lim = f(a)]
Right: “Need to verify: (1) f(2) exists, (2) lim exists, (3) they’re equal.”
Fix: Continuity requires all three conditions. Missing one means discontinuous.
Mistake 4: Misapplying Limit Laws
Example: “Find lim(x→2) [f(x)/g(x)].”
Wrong: “lim f / lim g” without checking that lim g ≠ 0.
Right: “Only if lim(x→2) g(x) ≠ 0. Otherwise the limit might not exist.”
Fix: Check the denominator limit first.
Mistake 5: Forgetting to Factor or Simplify Indeterminate Forms
Example: “Find lim(x→0) (3x² + 2x) / x.”
Wrong: “It’s 0/0, so no limit exists.”
Right: “Factor: x(3x + 2) / x = 3x + 2. So lim(x→0) (3x + 2) = 2.”
Fix: 0/0 means you haven’t simplified yet. Always factor, rationalize, or use algebraic tricks.
What to Do Next
To master basic limits: Evaluate 15 limits using direct substitution (continuous functions).
To master indeterminate forms: Resolve 15 limits of the form 0/0 by factoring.
To master continuity: Check continuity for 10 piecewise functions at boundary points.
The Problem Vault Alternative
The Problem Vault has 75+ Limits & Continuity problems from actual PUC exams.