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Section 2.4 To say a function is continuous at x = c means that there is NO interruption in the graph of f at c. The graph has no holes, gaps, or jumps. 1. The function is undefined at x = c 2. lim f ( x) DNE x c 3. lim f ( x) f (c) x c A function f is continuous at c IFF ALLare true… 1. f(c) is defined. f ( x) exists 2. lim x c 3. lim f ( x) f (c) x c A function is continuous on an interval (a, b) if it is continuous at each pt on the interval. A function is discontinuous at c if f is defined on (a, b) containing c (except maybe at c) and f is not continuous at c. 1. Removable : You can factor/cancel out, therefore making it continuous by redefining f(c). 2. Non-Removable: You can’t remove it/cancel it out! 1. Removable: x2 4 f ( x) x2 ( x 2)( x 2) x2 x2 We “removed” the (x-2). ▪ Therefore, we have a REMOVABLE DISCONTINUITY when x – 2 = 0, or, when x = 2. Non-Removable: f ( x) 1 x 1 We can’t remove/cancel out this discontinuity, so we have a NON-Removable discontinuity when x – 1 =0, or when x = 1. ▪ We will learn that Non-Removable Discontinuities are actually Vertical Asymptotes! 1. Set the deno = 0 and solve. 2. If you can factor and cancel out (ie-remove it) you have a REMOVABLE Discontinuity. 3. If not, you have a NON-Removable Discontinuity. You can evaluate limits for the left side, or from the right side. x approaches c from values that are greater than c. x approaches c from values that are less than c. 1. lim 3 x x 3 =0 2. lim x 0 lim x 0 x =1 x x = -1 x Therefore, the limit as x approaches 0 DNE!! 4 x, x 1 3. f ( x ) 2 4 x x , x 1 lim f ( x) 3 x 1 lim f ( x) 3 lim f ( x) 3 x 1 x 1 1. Factor and cancel as usual. 2. Evaluate the resulting function for the value when x=c. 3. If this answer is NOT UNDEFINED then that is your solution. 4. If this answer is UNDEFINED, then graph the function and look at the graph for when x=c. 2 x 3, x 1 f ( x); f ( x) 2 Ex: lim x 1 x , x 1 Evaluate each function separately for the value when x=c. If the solutions are all the same, that is your limit. If they are not, then the limit DNE. 2 x 3, x 1 lim f ( x); f ( x) 2 x 1 x , x 1 lim 2 x 3 x 1 lim x x 1 2 =1 =1 Therefore, the limit as x approaches 1 of f(x) =1