MATH U101 · Multivariable Calculus · Module 2

Limits, Continuity & Vector-Valued Functions

Two halves: the self-study revision block §2.3–2.6 (limits & continuity of single-variable functions), then the new material — vector functions, motion, arc length and curvature §13.1–13.4.

How to read this module strategically

The handout marks §2.3–2.6 as "self-study — revision of 12th class topics." This is the one block where BITSAT students genuinely have a head start, and the one block the lecturer will not teach — so this is exactly where your extra weekly hours should go, and go first. Part A below is a complete map of it. Part B (vector functions) is new to everyone and builds directly on Part A: a vector function is just three ordinary functions in a trench coat.

Part A · Limits and continuity, the complete refresher

Thomas §2.3–2.6 — self-study block. Budget: the biggest single slice of catch-up time.

A1 · What a limit is

lim_x→a f(x) = L says: as x gets close to a — from either side, without ever equalling a — the values f(x) get and stay close to L.

The crucial subtlety: the limit is about the journey, not the destination. f(a) may be undefined, or defined with a different value — the limit doesn't care. That's why (x²−9)/(x−3) has a perfectly good limit at x = 3 even though the function is undefined there.

The 0/0 toolbox — 90% of limit problems

If plugging in gives a number, that's the answer (for the nice functions in this course). If it gives 0/0, the form is hiding a common factor. Three tools, in order of frequency:

Tool	Use when	One-line example
Factor & cancel	Polynomials	lim_x→3 x²−9x−3 = lim_x→3 (x+3) = 6
Rationalise	Square roots	lim_x→0 √(x+1) − 1x = lim_x→0 1√(x+1) + 1 = ½
Known special limit	Trig over x	lim_θ→0 sin θθ = 1 (θ in radians — memorise)

A fourth tool worth knowing by name: the Sandwich (Squeeze) Theorem — if g ≤ f ≤ h near a and g and h share the limit L, then f → L too. Its signature use: lim_x→0 x² sin(1/x) = 0, squeezed between −x² and x².

A2 · The precise (ε–δ) definition §2.3

This is the part that is genuinely new even to BITSAT students — 12th boards state limits informally; Thomas §2.3 makes it rigorous. Think of it as a challenge game between a skeptic and you:

The game

Skeptic: "I bet f(x) doesn't really stay within ε = 0.01 of L."
You: "It does — keep x within δ = 0.003 of a and you'll see."
The limit exists if you can win for every ε the skeptic picks, by producing a suitable δ. Formally: for every ε > 0 there is a δ > 0 such that 0 < |x − a| < δ ⟹ |f(x) − L| < ε.

Worked example 1Prove lim_x→2 (3x − 1) = 5 with ε–δ

Start from what must be small: |f(x) − 5| = |3x − 6| = 3|x − 2|.
We need 3|x − 2| < ε, i.e. |x − 2| < ε/3.
So choose δ = ε/3. Then 0 < |x−2| < δ forces |f(x)−5| < ε. Done.

Every linear ε–δ proof is this same two-line algebra: express |f(x) − L| as (constant)·|x − a|, then divide ε by the constant. Master the linear case cold; anything harder is rarely examined at this level.

A3 · One-sided limits §2.4

x → a⁻ approaches from the left, x → a⁺ from the right. The two-sided limit exists iff both one-sided limits exist and agree. The canonical counterexample: f(x) = |x|/x, which is −1 for negative x and +1 for positive x — the one-sided limits at 0 are −1 and 1, so lim_x→0 does not exist.

One-sided limits are the working tool for every piecewise function — and piecewise functions are the exam's favourite way to test this whole block.

A4 · Continuity §2.5

f is continuous at a when three things hold — and exam answers are expected to check all three:

f(a) exists (the point is on the graph),
lim_x→a f(x) exists (both sides agree),
the two are equal.

Polynomials, sin, cos, exponentials are continuous everywhere; rational functions everywhere their denominator isn't zero; sums, products, quotients (nonzero denominator) and compositions of continuous functions are continuous. In practice: to evaluate a limit of a continuous function, just substitute.

Worked example 2Find c so that f is continuous everywhere

f(x) = x² for x < 1, and f(x) = 2x − c for x ≥ 1.

Each piece is a polynomial — continuous on its own territory. The only risk is the seam at x = 1.
Left limit: x² → 1. Right limit (and value): 2(1) − c = 2 − c.
Match them: 2 − c = 1 ⟹ c = 1.

This "match at the seam" pattern is the single most common exam question from §2.5.

One theorem to know by name: the Intermediate Value Theorem — a function continuous on [a, b] takes every value between f(a) and f(b). Standard use: "show x³ − x − 1 = 0 has a root in [1, 2]" — evaluate at both ends (−1 and 5), note the sign change, cite IVT.

A5 · Limits at infinity & asymptotes §2.6

For x → ±∞ in a ratio, divide top and bottom by the highest power of x in the denominator. The shortcut you'll use forever, for rational functions:

Degrees equal → limit is the ratio of leading coefficients: (3x²+5)/(2x²−x) → 3/2 (horizontal asymptote y = 3/2).
Denominator degree higher → limit 0.
Numerator degree higher → limit ±∞ (no horizontal asymptote; exactly one degree higher gives an oblique asymptote, found by long division).
Vertical asymptotes: where the denominator → 0 but the numerator doesn't. If both → 0, factor first — it may be a hole, not an asymptote.

Classic traps in Part A

1) Writing lim = f(a) by reflex when the function is piecewise or has a hole — check the seam. 2) Using sin x/x → 1 in degrees mode — it's a radians fact. 3) "Denominator is 0, so vertical asymptote" — not if the numerator is 0 too (factor first!). 4) Saying a limit "= ∞" and calling it "exists" — ∞ is a description of behaviour, not a number; the limit does not exist.

Part B · Vector-valued functions and motion in space

Thomas §13.1–13.4 — new to everyone. Prereqs: Part A, vectors, differentiation rules.

B1 · A curve as a moving point §13.1

A vector-valued function feeds in a time t and returns a position vector:

r(t) = f(t) i + g(t) j + h(t) k = ⟨f(t), g(t), h(t)⟩

As t runs, the tip of r(t) traces a curve in space. The mental model for the whole chapter: a fly buzzing along a path. The path is geometry; the fly's speed along it is dynamics — this chapter connects the two (exactly the "relation between dynamics and geometry of motion" your learning outcome promises).

The headline example, used in every section: the helix r(t) = ⟨cos t, sin t, t⟩ — a point circling the z-axis while rising at a steady rate, like a spiral staircase.

The one idea that makes this chapter easy

Everything happens componentwise. Limits, continuity, derivatives, integrals of r(t) are all computed by doing ordinary single-variable calculus to f, g, h separately. A vector function is three scalar functions in a trench coat. If you can differentiate cos t, you can differentiate a helix. Continuity of r at a = continuity of all three components at a — which is why Part A had to come first.

B2 · Velocity, speed, acceleration

Differentiate componentwise; the results have physical names:

Object	Definition	Type	Meaning
Velocity	v(t) = r′(t)	vector	tangent to the path, points where you're heading
Speed	\|v(t)\|	scalar	how fast, ignoring direction
Acceleration	a(t) = v′(t)	vector	how velocity changes (turning counts, even at constant speed)
Unit tangent	T = v/\|v\|	unit vector	pure direction of travel

The differentiation rules all survive, including two product rules — note the cross-product rule preserves order, because u × v ≠ v × u:

(u · v)′ = u′ · v + u · v′ (u × v)′ = u′ × v + u × v′

A beautiful consequence worth quoting in exams: if |r(t)| is constant (motion on a sphere or circle), then differentiating r · r = const gives r · r′ = 0 — position and velocity are perpendicular. That's why the moon's velocity is always at right angles to its position vector.

B3 · Arc length §13.3

Why this matters

Distance travelled = speed × time, integrated: over a tiny dt the fly covers |v| dt. Sum the pieces:

L = ∫_a^b |v(t)| dt = ∫_a^b √(f′(t)² + g′(t)² + h′(t)²) dt

The arc-length parameter s(t) is distance travelled since a start time; its key property is ds/dt = |v| ("rate of covering distance = speed"). It exists so curvature (next) can be defined in terms of the road, not the driver.

B4 · Curvature and the normal vector §13.4

Curvature κ measures how sharply the road bends, per metre of road. Sharp hairpin = large κ; motorway = small κ; straight line = 0. Definition: how fast the direction T turns per unit of distance (not time — bending is a property of the road, not of how fast you drive it):

κ = |dT/ds| = |dT/dt||v| N = dT/dt|dT/dt|

The principal unit normal N points the way the curve is turning — always perpendicular to T (because T has constant length 1, the perpendicularity fact from B2 applies). Anchor facts to sanity-check every answer: a straight line has κ = 0; a circle of radius a has κ = 1/a — the bigger the circle, the gentler the bend.

Worked example 3Full workout on the helix r(t) = ⟨cos t, sin t, t⟩

This one example exercises every formula in Part B — it is the exam rehearsal.

Velocity: differentiate componentwise: v = ⟨−sin t, cos t, 1⟩.
Speed: |v| = √(sin²t + cos²t + 1) = √2 — constant. (The identity sin²+cos² = 1 saves the day, as it will constantly.)
Arc length over one full turn 0 ≤ t ≤ 2π: L = ∫₀^2π √2 dt = 2π√2.
Unit tangent: T = v/√2 = ⟨−sin t, cos t, 1⟩/√2.
Turn rate: dT/dt = ⟨−cos t, −sin t, 0⟩/√2, so |dT/dt| = 1/√2.
Curvature: κ = (1/√2)/√2 = 1/2 — constant, and sensibly less than the κ = 1 of the unit circle it shadows: stretching a circle upward into a helix straightens it slightly.
Normal: N = ⟨−cos t, −sin t, 0⟩ — horizontal, pointing straight at the z-axis. The helix is forever "turning toward" its own axis. Check: T · N = 0 ✓.

Classic traps in Part B

1) Speed is not velocity — one is a scalar, one a vector; exam questions dock marks for answering with the wrong type. 2) |r′(t)| ≠ (|r(t)|)′ — the magnitude of the derivative is not the derivative of the magnitude. Compute r′ first, then take the magnitude. 3) Zero acceleration is not implied by constant speed — the helix has constant speed √2 yet nonzero acceleration, because direction changes. 4) Forgetting to normalise: T and N must have length 1 — always divide by the magnitude and verify.

Your minimal prerequisite kit for this module

Differentiation rules, fluent: power, product, quotient, chain; derivatives of sin, cos, e^x, ln x, polynomials. This is the BITSAT kids' real edge — close it with drilling (20 minutes a day for two weeks beats a weekend binge).
Basic integration: power rule, sin, cos, e^x; definite integrals. Needed for arc length.
Vectors: components, magnitude |v| = √(v₁²+v₂²+v₃²), unit vectors, dot product (and that u · v = 0 ⟺ perpendicular), cross product basics. If shaky, skim Thomas §12.1–12.4 — an evening's read.
Algebra hygiene: factoring, rationalising, |x| as a piecewise function.
sin²t + cos²t = 1 — reached for constantly in §13.

What to practise in Thomas

Skill	Where	How many
0/0 limits (factor, rationalise, sin θ/θ), one-sided limits	§2.4 exercises	10–12
ε–δ for linear functions	§2.3 exercises	3–4
Continuity of piecewise functions, find-the-constant, IVT	§2.5 exercises	6–8
Limits at ∞, all asymptote types	§2.6 exercises	6–8
r(t): sketch, v, a, speed	§13.1 exercises	6–8
Arc length; T	§13.3 exercises	4–6
Curvature, N (circles, helices, plane curves)	§13.4 exercises	4–6

(§13.2 — integrals of vector functions and projectile motion — is starred on the handout; skim the integral definition, and ask the instructor whether projectile motion is examinable before investing time there.)