Limits lim x→a f(x) = L lim x→4 2x+3 = 11 x y 3.99 3.6 4 4.4 4.1 10.2 Limit: A function f has the limit L as x approaches a, written: If all values f(x) are close to L for values of x that are arbitrarily close to but not equal to a itself. lim x→a f(x) = L Example: Find the limit of f(x) = 2x+3 when x=4 lim x→4 2x+3 = 11 3 4 5 1 10 x y 3.99 3.6 4 4.4 4.1 10.2 11 11.2 11.8 10.98

Rules of Limits lim x→a f(x) = L lim x→a- f(x) = L Continuity In order for a limit to exist, both the left and right values MUST exist and be the same value L. lim x→a f(x) = L lim x→a- f(x) = L From the LEFT From the RIGHT Whether or not a limit exists at a has nothing to do with the function value f(a). Continuity When the limit value of a function is the came as the function value, it satisfies a condition called Continuity at a point. If a function is continuous, then there are no jumps., or holes in the graph and a pencil can draw the function without being lifted off the paper.

Four Principles of Continuity A function f is continuous at x = a if: f(a) exists limx→a f(x) exists and limx→a f(x)=f(a) A function f is continuous over an interval I if it is continuous at each point in I. Four Principles of Continuity Any Constant function is continuous, since such a function never varies. For any positive integer n, and any continuous function f, [f(x)]n and n√f(x) are continuous. --When n is even the solutions to n√f(x) are restricted to f(x)≥0. If f(x) and g(x) are both continuous functions, then so is their sum, difference and product. If f(x) and g(x) are both continuous functions, then so is their quotient as long as the denominator does not equal zero.

Properties of Limits