Module 1: Limits and Continuity
Calculus I – Study Guide
Learn Without Walls
1. Limits
Informal Definition
limx→a f(x) = L means f(x) gets arbitrarily close to L as x approaches a (from both sides), regardless of the value of f(a).
One-Sided Limits
- Left-hand: limx→a− f(x) — approach from x < a
- Right-hand: limx→a+ f(x) — approach from x > a
- Two-sided limit exists if and only if both one-sided limits exist and are equal.
Limit Laws
| Law | Statement |
|---|---|
| Sum | lim [f + g] = lim f + lim g |
| Difference | lim [f − g] = lim f − lim g |
| Product | lim [f · g] = (lim f)(lim g) |
| Quotient | lim [f/g] = (lim f)/(lim g), g ≠ 0 |
| Power | lim [f]n = (lim f)n |
| Constant Multiple | lim [cf] = c · lim f |
Algebraic Techniques
- Direct substitution — try first; works if no 0/0.
- Factor and cancel — factor numerator/denominator, cancel common factor.
- Rationalize — multiply by conjugate to clear radicals.
- Trig identities — use (sin x)/x → 1 and (1 − cos x)/x → 0.
- Squeeze Theorem — trap f between two functions with the same limit.
Special Limits:
limx→0 (sin x)/x = 1
limx→0 (1 − cos x)/x = 0
limx→0 (sin x)/x = 1
limx→0 (1 − cos x)/x = 0
2. Limits at Infinity
Rational Functions at Infinity
Compare degrees of numerator (n) and denominator (m):
n < m: limit = 0 | n = m: limit = an/bm | n > m: limit = ±∞
n < m: limit = 0 | n = m: limit = an/bm | n > m: limit = ±∞
Asymptotes
| Type | How to Find |
|---|---|
| Horizontal (y = L) | limx→±∞ f(x) = L |
| Vertical (x = a) | Denominator = 0 while numerator ≠ 0 |
3. Continuity
Three Conditions
f is continuous at x = a if: (1) f(a) is defined, (2) limx→a f(x) exists, (3) limx→a f(x) = f(a).
Types of Discontinuities
| Type | Description |
|---|---|
| Removable | Limit exists but f(a) is wrong or missing (hole) |
| Jump | One-sided limits both exist but differ |
| Infinite | At least one side → ±∞ (vertical asymptote) |
Families of Continuous Functions
Polynomials, rational functions (on domain), trig functions, exponentials, logarithms, and compositions/sums/products of these are continuous on their domains.
4. Intermediate Value Theorem
If f is continuous on [a, b] and N is between f(a) and f(b), then there exists c in (a, b) with f(c) = N.
Using the IVT
- Verify f is continuous on [a, b].
- Compute f(a) and f(b).
- Show that the target value N lies between f(a) and f(b).
- Conclude that a solution exists in (a, b).
Example: Show x³ − x − 1 = 0 has a root in [1, 2].
f(1) = −1 < 0, f(2) = 5 > 0. Continuous polynomial, sign change ⇒ root exists.
f(1) = −1 < 0, f(2) = 5 > 0. Continuous polynomial, sign change ⇒ root exists.