Introduction to Limits

Grade 12 · Math · Study Guide

Understand the foundational concept of limits and how they describe function behavior as inputs approach a value.

A limit describes what a function's output approaches as the input gets closer and closer to a specific value — without necessarily reaching it. Written as lim(x→c) f(x) = L, this reads: 'as x approaches c, f(x) approaches L.' The key insight is that the limit is about the journey, not the destination.

Consider f(x) = (x² - 1)/(x - 1). At x = 1, we'd be dividing by zero — the function is undefined there. But factor the numerator: (x-1)(x+1)/(x-1) = x+1 for all x ≠ 1. As x approaches 1, f(x) approaches 2. The function has a 'hole' at x = 1, but the limit still exists and equals 2.

Limits can be one-sided: lim(x→c⁺) means approaching from the right, lim(x→c⁻) from the left. A two-sided limit exists only when both one-sided limits are equal. Limits underpin the entire calculus — derivatives are defined as limits of difference quotients, and integrals use limits of Riemann sums. Mastering limit intuition is the gateway to all of higher mathematics.

Key terms

Limit: The value a function approaches as the input gets arbitrarily close to a given value.
One-Sided Limit: A limit where x approaches c from only one direction — left (c⁻) or right (c⁺).
Indeterminate Form: An expression like 0/0 or ∞/∞ that requires algebraic manipulation or L'Hôpital's Rule to evaluate.
Continuity: A function is continuous at c if the limit as x→c exists and equals f(c).
Limit Laws: Rules allowing limits to be computed using sums, products, and quotients of simpler limits.

Flashcards

lim(x→c) f(x) = L: As x approaches c from either side, f(x) gets arbitrarily close to L.
One-Sided Limit: lim(x→c⁺) approaches from the right; lim(x→c⁻) from the left.
Two-Sided Limit: Exists only when the left-hand and right-hand limits are equal.
Indeterminate Form: Forms like 0/0 or ∞/∞ that signal the need for further algebraic work.
Continuity at a Point: f is continuous at c if lim(x→c) f(x) exists, f(c) is defined, and they are equal.
Limit Laws: Sum, difference, product, quotient, and constant-multiple rules for computing limits.
Hole (Removable Discontinuity): A single missing point in a function's graph where the limit exists but f(c) is undefined or wrong.
Squeeze Theorem: If g(x) ≤ f(x) ≤ h(x) and g and h share the same limit L at c, then f's limit is also L.

Questions

What is lim(x→3) (x² - 9)/(x - 3)?
1. 0
2. 3
3. 6
4. undefined
Show answer
Answer: 6
Factor: (x²-9)/(x-3) = (x-3)(x+3)/(x-3) = x+3 for x≠3. As x→3, x+3→6.
A limit can exist at a point even if the function is undefined there.
Show answer
Answer: True
The limit depends on surrounding behavior, not the function value at the point itself. f(x) = (x²-1)/(x-1) is undefined at x=1 but has limit 2.
If lim(x→2⁻) f(x) = 5 and lim(x→2⁺) f(x) = 7, what can we conclude?
1. The limit equals 5
2. The limit equals 7
3. The limit equals 6
4. The two-sided limit does not exist
Show answer
Answer: The two-sided limit does not exist
For a two-sided limit to exist, both one-sided limits must agree. Since 5 ≠ 7, the two-sided limit does not exist.
The expression 0/0 encountered when evaluating a limit is called a(n) ___ form.
Show answer
Answer: indeterminate
0/0 is an indeterminate form — algebraic manipulation (factoring, conjugates) or L'Hôpital's Rule is needed to find the actual limit.
Which condition is required for f(x) to be continuous at x = c?
1. f(c) must be defined only
2. The limit must exist only
3. Both the limit and f(c) must exist and be equal
4. The function must be differentiable at c
Show answer
Answer: Both the limit and f(c) must exist and be equal
Continuity at c requires three things: f(c) is defined, the limit exists, and f(c) equals the limit. All three must hold simultaneously.

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