The core idea: A limit asks: what value does f(x) get closer and closer to as x approaches a? Write: lim as x → a of f(x) = L Read: "the limit of f(x) as x tends to a equals L" The limit does NOT care what happens AT x = a — only what happens NEAR it.
Think of walking towards a door — the limit is the door, even if you never actually open it. The function might be undefined at x = a (a hole in the graph), but the limit can still exist.
Common mistake: Students confuse the limit with the function value. They are different! • f(a) = value AT x = a (may not exist) • lim as x → a of f(x) = value the function APPROACHES near x = a A limit can exist even when f(a) does not.
IB exam tip: In Paper 1 and Paper 2, limits appear in: • Evaluating limits by substitution (show working) • Factorising to remove a 0/0 form • Reading limits from a graph or table Always show the substitution step — do not just write the answer.
What is an indeterminate form?: When you substitute x = a and get 0/0, the problem is NOT solved. This is called an indeterminate form — the limit might exist, but you cannot find it by substitution alone. You must simplify first (factor, cancel, use conjugate, or rewrite the expre ion). Only after simplifying can you substitute x = a.
Strategy: factor and cancel: Step 1: Try direct substitution. Step 2: If you get 0/0, factorise the numerator (or denominator). Step 3: Cancel the common factor. Step 4: Substitute again.
0/0 form (need to factorise)
- (x2 - 9)/(x - 3) at x = 3
- (x2 - 1)/(x - 1) at x = 1
- (x2 - 4)/(x - 2) at x = 2
After factorising and cancelling
- x + 3 → limit = 6
- x + 1 → limit = 2
- x + 2 → limit = 4
Worked example
Apply the key method from Introduction to Limits in a typical IB-style question.
Step by step
- Write the relevant formula or rule first.
- Substitute values carefully and show each step.
- State the final answer with correct units/context.
Final answer
Clear method and context-based interpretation secure most marks.
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Two ways to approach a point: For any point x = a, you can approach from the left (smaller values) or from the right (larger values). • Left-hand limit: lim as x → a− of f(x) — approach from values below a • Right-hand limit: lim as x → a+ of f(x) — approach from values above a The two-sided limit (lim as x → a of f(x)) only exists if BOTH one-sided limits are equal.
The existence rule: lim as x → a of f(x) = L if and only if: lim as x → a− of f(x) = L AND lim as x → a+ of f(x) = L If left ≠ right, the two-sided limit does NOT exist (DNE).
When to expect a one-sided question: In IB paper, one-sided limits usually appear with: • Piecewise functions (different rules for different intervals) • Graphs with a jump discontinuity • Tables asking you to approach from two side Always check both sides before concluding the limit exist.
Limit EXISTS
- Left limit = Right limit
- Both sides approach the same value
- f(a) can be anything (or undefined)
- Example: limx → 2 f(x) = 5
Limit does NOT EXIST
- Left limit ≠ Right limit
- Sides approach different values
- Called a jump discontinuity
- Write: limit DNE
How to write this in exams: If asked to show a limit does not exist: 1. Find the left-hand limit and state the value 2. Find the right-hand limit and state the value 3. State: since left ≠ right, the two-sided limit does not exist Do NOT just write "DNE" without evidence — show both side.
Worked example
Apply the key method from Introduction to Limits in a typical IB-style question.
Step by step
- Write the relevant formula or rule first.
- Substitute values carefully and show each step.
- State the final answer with correct units/context.
Final answer
Clear method and context-based interpretation secure most marks.
IB exams regularly test limits from tables of values and graph. These do not require any algebra — only careful reading. This section is high-value because these questions appear on both Paper 1 and Paper 2.
Reading a limit from a table: A table shows x-values getting closer to a from both side. Look at the f(x) column — what value does it approach? Example table for lim as x → 2 of f(x): x: 1.9 | 1.99 | 1.999 | 2.001 | 2.01 | 2.1 f(x): 3.81 | 3.98 | 3.999 | 4.001 | 4.02 | 4.21 Both sides approach 4, so the limit is 4.
Reading a limit from a graph: Trace the graph from the LEFT towards x = a — what y-value does it head toward? Trace from the RIGHT — what y-value does it head toward? If both approaches reach the SAME y-value, that is the limit. Look for: open circles (hole — function undefined there), closed circles (function defined), or jump.
The three limit scenarios in IB graphs: 1. Open circle at (a, L): limit = L, but f(a) is undefined 2. Closed circle at (a, L): limit = L and f(a) = L (continuous) 3. Jump: closed circle at (a, L1) approaching from one side, open circle at (a, L2) from other — limit DNE
IB graph questions — how to score full marks: When asked for a limit from a graph: 1. State the left-hand limit value explicitly 2. State the right-hand limit value explicitly 3. If equal: write the limit 4. If not equal: write 'limit does not exist' Do not just point at the graph — write value.