Continuity, asymptotes and the intermediate value theorem

Learning objectives

Classify a discontinuity as removable, jump or infinite (or oscillating) from a graph, a table or a formula, using one-sided limits to decide which type it is

State the three conditions for continuity at a point (f(a) exists, the limit exists, and the two are equal) and use them to test continuity of piecewise-defined functions, including finding parameter values that make a function continuous

Confirm continuity over an interval, using the fact that polynomial, rational, power, exponential, logarithmic and trigonometric functions are continuous on their domains, and remove a removable discontinuity by redefining the function at one point

Connect infinite limits to vertical asymptotes, determining the sign of each one-sided limit near an asymptote from the factors of a rational function

Connect limits at infinity to horizontal asymptotes and end behavior, comparing the growth rates of polynomial, exponential and logarithmic terms in the numerator and denominator

Apply the intermediate value theorem to justify that a continuous function takes a given value or has a zero on a closed interval, stating the hypotheses that must be checked