Limits and Continuity

Unit Overview

Limits are the foundation on which all of calculus is built. Before we can talk about instantaneous rates of change (derivatives) or accumulated change (integrals), we need a precise way to describe what a function approaches as its input gets arbitrarily close to some value. That idea — the limit — is what this unit develops.

By the end of this unit you should be able to reason about limits four ways: graphically, numerically, algebraically, and verbally, and connect them to the idea of continuity. On the AP exam, roughly 10–12% of the questions come from this unit, and the ideas reappear everywhere afterward.

Big picture: A limit answers the question, "Where is the function heading?" — regardless of whether it ever actually arrives.

Learning Objectives

After completing this unit you will be able to:

• Distinguish average rate of change (slope of a secant line) from instantaneous rate of change (slope of a tangent line).
• Express and evaluate limits using correct notation, including one-sided limits.
• Estimate limits from graphs and tables.
• Evaluate limits algebraically using limit laws, factoring, rationalization, and known trigonometric limits.
• Apply the Squeeze Theorem.
• Classify discontinuities (removable, jump, infinite) and remove removable ones.
• Determine continuity at a point and over an interval.
• Connect infinite limits to vertical asymptotes and limits at infinity to horizontal asymptotes.
• Apply the Intermediate Value Theorem (IVT).

Prerequisite Concepts

You should be comfortable with:

• Function notation and evaluation, domain and range.
• Factoring (difference of squares, difference of cubes, common factors) and rationalizing with conjugates.
• Piecewise functions and reading graphs.
• Basic trigonometry (unit-circle values, identities like tan = sin/cos).
• Rational functions and asymptote behavior from precalculus.

Key Definitions

• Limit: We write lim_{x→c} f(x) = L if the values of f(x) get arbitrarily close to L as x gets arbitrarily close to c (from both sides), without requiring x = c.
• One-sided limits: lim_{x→c⁻} f(x) (from the left) and lim_{x→c⁺} f(x) (from the right).
• Existence of a two-sided limit: lim_{x→c} f(x) exists iff both one-sided limits exist and are equal.
• Average rate of change of f on [a, b]: (f(b) − f(a)) / (b − a) — the slope of the secant line.
• Instantaneous rate of change at x = a: lim_{h→0} (f(a + h) − f(a)) / h — the slope of the tangent line.
• Continuity at x = a: f is continuous at a if all three hold: (1) f(a) is defined, (2) lim_{x→a} f(x) exists, (3) lim_{x→a} f(x) = f(a).
• Removable discontinuity: a "hole" — the limit exists but does not equal f(a) (or f(a) is undefined).
• Jump discontinuity: the one-sided limits both exist but are unequal.
• Infinite discontinuity: the function grows without bound near the point (a vertical asymptote).

Important Theorems

Limit Laws (valid when the individual limits exist). If lim f = L and lim g = M:

• Sum/Difference: lim (f ± g) = L ± M
• Constant multiple: lim (k·f) = k·L
• Product: lim (f·g) = L·M
• Quotient: lim (f/g) = L/M, provided M ≠ 0
• Power/Root: lim (f)^n = L^n

Squeeze (Sandwich) Theorem: If g(x) ≤ f(x) ≤ h(x) near c (except possibly at c) and lim_{x→c} g(x) = lim_{x→c} h(x) = L, then lim_{x→c} f(x) = L.

Intermediate Value Theorem (IVT): If f is continuous on [a, b] and N is any value between f(a) and f(b), then there exists at least one c in (a, b) with f(c) = N. (Guarantees existence, not the value or uniqueness.)

Continuity of common functions: Polynomials are continuous everywhere; rational, radical, trig, exponential, and log functions are continuous on their domains; sums, products, quotients (nonzero denominator), and compositions of continuous functions are continuous.

Important Formulas & Standard Limits

• Direct substitution (when f is continuous at c): lim_{x→c} f(x) = f(c).
• Fundamental trig limits:
  • lim_{x→0} (sin x)/x = 1
  • lim_{x→0} (1 − cos x)/x = 0
  • lim_{x→0} (tan x)/x = 1
• Limits at infinity of rational functions (compare degrees of numerator N and denominator D):
  • deg N < deg D → limit is 0 (horizontal asymptote y = 0)
  • deg N = deg D → limit is the ratio of leading coefficients
  • deg N > deg D → no horizontal asymptote (limit is ±∞)

Concept Explanations

From average to instantaneous rate (1.1). The average rate of change over [a, b] is a secant slope you can compute directly. To get the instantaneous rate at a single point, shrink the interval: let the second point slide toward the first (h → 0). That limiting secant slope is the tangent slope — the derivative you'll define formally in Unit 2.

Why a limit ignores f(c) (1.2). The whole power of a limit is that it describes approach, not arrival. A function can have a limit of 3 at x = 4 while f(4) = 5, or while f(4) is undefined. This is exactly what lets us handle the 0/0 forms that derivatives produce.

Algebraic techniques (1.5–1.6). Try direct substitution first. If you get a real number, you're done. If you get 0/0 (indeterminate), do algebra: factor and cancel, rationalize with a conjugate, or use a known trig limit. If you get (nonzero)/0, the limit is infinite (check signs for ±∞).

Discontinuities and continuity (1.10–1.13). Test continuity with the three-part definition. A removable discontinuity can be "patched" by redefining f(c) to equal the limit. Jump and infinite discontinuities cannot be removed.

Asymptotes as limits (1.14–1.15). A vertical asymptote at x = c means a one-sided (or two-sided) limit is ±∞. A horizontal asymptote y = L means lim_{x→±∞} f(x) = L.

Worked Examples

Example 1 (Algebraic — factoring). Evaluate lim_{x→3} (x² − 9)/(x − 3). Direct substitution gives 0/0. Factor: (x − 3)(x + 3)/(x − 3) = x + 3. So the limit is 3 + 3 = 6.

Example 2 (Rationalizing). Evaluate lim_{x→0} (√(x + 1) − 1)/x. Multiply by the conjugate: ((x + 1) − 1) / (x(√(x+1) + 1)) = 1/(√(x+1) + 1). As x → 0 this is 1/(1 + 1) = 1/2.

Example 3 (Squeeze Theorem). Evaluate lim_{x→0} x² sin(1/x). Since −1 ≤ sin(1/x) ≤ 1, we have −x² ≤ x² sin(1/x) ≤ x². Both bounds → 0, so the limit is 0.

Example 4 (Continuity — solve for a constant). Find k so that f is continuous at x = 2, where f(x) = kx² for x ≤ 2 and f(x) = x + k for x > 2. Match the pieces: k(2)² = 2 + k → 4k = 2 + k → 3k = 2 → k = 2/3.

Example 5 (Limit at infinity). Evaluate lim_{x→∞} (3x² + 5)/(2x² − x). Equal degrees → ratio of leading coefficients = 3/2.

Example 6 (IVT). Show f(x) = x³ − x − 1 has a root in [1, 2]. f is continuous; f(1) = −1 < 0 and f(2) = 5 > 0. Since 0 lies between −1 and 5, IVT guarantees a c in (1, 2) with f(c) = 0.

Graphical Interpretation

• A two-sided limit exists when the left and right branches of the graph head to the same height — even if there's a hole or the point is plotted elsewhere.
• A hole at (c, L) with the curve approaching L from both sides ⇒ lim = L (removable).
• A jump (left branch heads to one height, right branch to another) ⇒ two-sided limit does not exist.
• A vertical asymptote ⇒ the branch shoots to ±∞ (infinite limit).
• A horizontal asymptote ⇒ the far-left/far-right tails flatten toward y = L.

Common Mistakes

• Assuming lim_{x→c} f(x) = f(c) always. It only holds when f is continuous at c.
• Stopping at 0/0 and writing "does not exist." 0/0 is indeterminate — do more algebra.
• Confusing "limit does not exist" with "= ∞". An infinite limit is a specific behavior; a jump is a different reason a limit fails.
• Forgetting to check both one-sided limits for piecewise functions and at asymptotes.
• Sign errors on infinite limits. Near a vertical asymptote, test the sign from each side to decide +∞ vs −∞.
• Misusing IVT. It requires continuity and only guarantees existence — never a specific value or uniqueness.
• Canceling a factor and forgetting the hole. After canceling (x − c), the original function still has a removable discontinuity at x = c.

AP Exam Tips

• Always try direct substitution first — it settles most non-calculator limits instantly.
• Memorize the three fundamental trig limits; they appear on both multiple-choice and free-response.
• For rational-function end behavior, compare degrees rather than doing long division under time pressure.
• On free-response, when citing IVT or discussing continuity, state that the function is continuous — that justification earns points.
• Read whether a question asks for the limit or the function value — they can differ.
• Calculator-active tables: use x-values very close on both sides to estimate a limit.

Unit Summary

• A limit describes where a function heads as x → c, independent of f(c).
• A two-sided limit exists iff both one-sided limits exist and are equal.
• Evaluate limits by substitution → algebra (factor/rationalize/trig) → interpret ∞.
• Continuity requires the value, the limit, and their equality; classify breaks as removable, jump, or infinite.
• Infinite limits ⇔ vertical asymptotes; limits at infinity ⇔ horizontal asymptotes.
• The Squeeze Theorem handles oscillating bounded functions; the IVT guarantees solutions for continuous functions.

Next unit: Unit 2 turns the instantaneous-rate idea into the formal derivative.