AP CALCULUS AB REVIEW OF LIMITS

To Find a Limit Algebraically To evaluate a limit algebraically as x approaches a finite number c, substitute c into the expression. This is called direct substitution. (1) If the answer is a finite, that is the limit. (2) If the answer is of the form 0/0 we have an indeterminate form. In this case you need to simplify the expression by trying: (i) Factor numerator/denominator, simplify, then substitute c. (ii) Rationalize the numerator/denominator using the conjugate, simplify, then substitute c. (iii) Simplify the complex fraction, then substitute c. [Exs] See White Board For Smith’s Examples.

Evaluating Limits Graphically If we are given the graph of a function we can often find the limit of the function at a particular value of c by looking at the graph. In order for the limit to exist at x = c we need the left and right hand limits to agree at x = c. Sometimes a table of values may be used to approximate the limit as x approaches c. [Ex] See White Board For Examples.

Limits as x  ∞ The limit of a rational function as x  ± ∞ is the value of the horizontal asymptote(s). If a function is not rational a table of values may be used. For more complicated rational functions we can divide numerator and denominator by the x-term with the largest power, then evaluate. Recall:_______________________ To prove a function has a horizontal asymptote you evaluate the limit of the function as x  ± ∞ [Exs] See White Board.

Special Limits The following limits involving sine and cosine are important: [Exs] See White Board

Infinite Limits Sometimes a function approaches ± ∞ as x  c. We call these infinite limits, but in fact the limit doesn’t exist. To find algebraically: (i)Simplify the top and bottom of the expression until the top is a constant and the bottom is 0. (ii)Judge if the denominator goes to 0 through positive values or negative values. (iii) Let c be a positive constant then:_____________________________ [Exs] See White Board

Continuity A function f(x) is continuous at x=c if: (i) f(c) is defined (ii) The limit as x  c of f(x) exists. (iii) The value of the limit = f(c) This is the definition we use to prove that a function is continuous at a point. If a function is continuous at every point in an interval then it is continuous on the interval.