Topic 2 math 256

What may possibly come out of a function is called the:

Codomain

Definition of Limit

We write lim(x->a) f(x) = L if f(x) approaches L as x approaches a.

Definition of Continuity on an Interval [a, b]

A function is continuous on an interval [a, b] if lim(x->a) f(x) = f(a).

Are all polynomials continuous?

Yes, all polynomials are continuous everywhere.

Squeeze Theorem

Evaluate lim(x->0) x * sin(1/x) using Squeeze Theorem

0 (Since -|x|

Evaluate lim(x->0) f(x) where f(x) = x+1 for x < 0 and f(x) = x+3 for x >= 0

DNE (Left-hand limit = 1, Right-hand limit = 3)

Express f(x) = |x| as a piecewise function

f(x) = x if x >= 0; f(x) = -x if x < 0

Evaluate lim(x->0) x / |x|

DNE (Piecewise: -1 for x < 0, 1 for x > 0; Left limit = -1, Right limit = 1)

Evaluate lim(x->1) 1 / (x - 1)^2

Infinity (+inf)

Definition of Continuity at a point x = a

A function is continuous at x = a if lim(x->a) f(x) = f(a).

When does a 2-sided limit exist?

Exactly when the left-hand limit (-) and right-hand limit (+) both exist and are equal.

Jump Discontinuity

Occurs when both left and right-hand limits exist, but they are not equal: lim(x->a-) f(x) != lim(x->a+) f(x).

Removable Discontinuity

Occurs when the limit lim(x->a) f(x) exists, but f(a) is undefined or f(a) != limit.

Infinite Discontinuity

Occurs when the function approaches positive or negative infinity from one or both sides (vertical asymptote).

6 main techniques for evaluating limits

1. Graphs & Tables, 2. Direct Substitution, 3. Limit Laws, 4. Factor & Cancel, 5. Common Denominators, 6. Conjugate / Rationalization

Limit Law: Sum and Difference lim(x->a) [f(x) +/- g(x)]

lim(x->a) f(x) +/- lim(x->a) g(x) = L +/- M

Limit Law: Constant Multiple lim(x->a) [c * f(x)]

c * lim(x->a) f(x) = c * L

Limit Law: Product lim(x->a) [f(x) * g(x)]

lim(x->a) f(x) * lim(x->a) g(x) = L * M

Limit Law: Quotient lim(x->a) [f(x) / g(x)]

lim(x->a) f(x) / lim(x->a) g(x) = L / M (provided M != 0)

What is the algebraic conjugate of a + b*sqrt(f(x))?

a - b*sqrt(f(x))