Calculus Intro Review

A function is continuous at a point if the following three conditions are met: 1) The function is defined at that point, 2) The limit of the function as it approaches that point exists, 3) The limit equals the function's value at that point.

A limit does not exist at a point if the function approaches different values from the left and right sides, or if it approaches infinity.

The limit of a function at a point is the value that the function approaches as the input approaches that point.

To determine continuity at a point, check if the function is defined at that point, if the limit exists, and if the limit equals the function's value.

A removable discontinuity occurs when a function is not defined at a point but can be made continuous by defining it appropriately. A non-removable discontinuity cannot be fixed by redefining the function at that point.

The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b], then it takes every value between f(a) and f(b) at least once.

A vertical asymptote is a line x = a where a function approaches infinity or negative infinity as the input approaches a.

A horizontal asymptote is a line y = b that a function approaches as the input approaches infinity or negative infinity.

The Squeeze Theorem states that if f(x) ≤ g(x) ≤ h(x) for all x in some interval, and if the limits of f(x) and h(x) both equal L as x approaches a, then the limit of g(x) as x approaches a also equals L.

The derivative of a function at a point is the limit of the average rate of change of the function as the interval approaches zero.