A limit describes the value that a function approaches as the input approaches some value, even if the function isn't defined there.

If a function is 'sandwiched' between two other functions that approach the same limit, then the function in the middle also approaches that limit.

The function must be defined at the point, the limit must exist at the point, and the limit must equal the function's value at the point.

Compare the degrees of the numerator and denominator. If equal, the asymptote is at the ratio of leading coefficients. If the denominator's degree is larger, the asymptote is at y=0.

If a function is continuous on a closed interval [a, b], it must take on every value between f(a) and f(b) at least once within that interval.

If $g(x) \leq f(x) \leq h(x)$ for all x near a, and $\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} f(x) = L$.

If f is continuous on [a, b] and k is any number between f(a) and f(b), then there exists at least one number c in [a, b] such that f(c) = k.

It indicates a removable discontinuity. The limit may exist even though the function is not defined at that point.

It indicates that the limit as x approaches that value is infinite or does not exist.

The graph will have a sudden 'jump' in value at a particular x-value, with different one-sided limits.