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Limits. 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. lim x→4 2x+3 = 11. Example: Find the limit of f (x) = 2x+3 when x=4. 10.
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Limits 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→af(x) = L lim x→4 2x+3 = 11 Example: Find the limit of f(x) = 2x+3 when x=4 10 x y 3.6 10.2 3.99 10.98 4 11 5 4.1 11.2 4.4 11.8 1 3 4 5
Rules of Limits In order for a limit to exist, both the left and right values MUST exist and be the same value L. From the LEFT From the RIGHT lim x→a- f(x) = L lim x→af(x) = L 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, thenthere are no jumps., or holes in the graph and a pencil can draw the function without being lifted off the paper.
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.
