1.6 One-SidedLimits&Continuity

1.6 One-SidedLimits&Continuity - 1.6 One-Sided...

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1.6 One-Sided Limits & Continuity One-sided limits: If f(x) approaches L as x tends toward c from the left (x < c), then lim ( ) x c f x L - = . If f(x) approaches M as x tends toward c from the right ( c < x ), then lim ( ) x c f x M + = . Continuity: A continuous function has no holes or gaps. To ensure that f(x) does not have a hole or gap at x = c ; the function must be defined at x = c , it must have a finite, two-sided limit at x = c ; and ( 29 lim x c f x must equal f(c) . If f(x) is not continuous at c , it is said to have a discontinuity there. Existence of a limit: The two-sided limit ( 29 lim x c f x exists if and only if the two one-sided limits exist and are equal: ( 29 ( 29 ( 29 lim lim lim x c x c x c f x f x f x - + = = . A polynomial or rational function is continuous wherever it is defined. Continuity on an interval: A function f(x) is continuous on an open interval a < x < b if it is continuous at each point x = c in that interval; it is continuous on the closed interval a x n if it is continuous on the open interval and ( 29 ( 29 lim x a f x f a + = and ( 29 ( 29 lim x b f x f b - = . Use the graphs to find the one-sided limits ( 29 2 lim x f x - and ( 29
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