Unit 1: Limits and Continuity (issue with rendering)

Limit

Describes what value a function’s outputs are approaching as the inputs get close to some number.

Two-sided limit

Exists when the left-hand limit and right-hand limit at a point are equal.

One-sided limit

Limits evaluated from only one side: left-hand limit (approaching a from the left) or right-hand limit (approaching a from the right).

Continuous function

A function is continuous at x = c if f(c) exists, the limit as x approaches c exists, and the limit equals f(c).

Removable discontinuity

Occurs when the limit exists but the function value is missing; can be removed by redefining the function at that point.

Jump discontinuity

Occurs when the left-hand and right-hand limits exist but are not equal.

Infinite discontinuity

Occurs when the function approaches infinity or negative infinity near a point, typically due to a vertical asymptote.

Squeeze Theorem

If g(x) ≤ f(x) ≤ h(x) and both g(x) and h(x) approach L as x approaches a, then f(x) also approaches L.

Horizontal asymptote

The line y = L represents end behavior; exists if lim x→∞ f(x) equals L.

Vertical asymptote

Occurs when f(x) approaches infinity or negative infinity as x approaches a from either side.

Intermediate Value Theorem (IVT)

If f is continuous on [a, b] and N is between f(a) and f(b), there exists c in [a, b] such that f(c) = N.

Indeterminate form

An expression like 0/0 that does not provide a specific limit value, indicating a need for further investigation.

Limit notation

$\lim_{x \to a} f(x) = L$ indicates that as x approaches a, f(x) approaches L.

Limit laws

Properties that allow the combination of limits, such as sum, difference, product, and quotient laws.

Fundamental Sine Limit

$\lim_{x \to 0} \frac{\sin(x)}{x} = 1$, a key limit when x is in radians.

Continuous on an interval

A function is continuous on [a, b] if it is continuous at every point in (a, b), and at the endpoints.

End behavior

Refers to the behavior of a function as x approaches infinity or negative infinity.

Direct substitution

Evaluating a limit by plugging in the value x approaches, applicable when no indeterminate form occurs.

Common mistake in limits

Assuming the limit exists without verifying left-hand and right-hand behaviors.

Limit noting DNE

DNE stands for 'does not exist'; often used when limits approach different values from either side.