Limits of Functions and Continuity

|a|a |x1|x1 |x2|x2 f (a) = L |a|a f(a) ≠ L o The Limit of a Function The limit as x approaches a (x → a) of f (x) = L means that as x gets closer and closer to a (on either side of a), f (x) must approach L. Here, f (a) does not need to exist for the limit to exist. f (x 1 ) f (x 2 ) f (x 1 ) f (x 2 ) |x1|x1 |x2|x2    

|a|a |x|x |x|x   L1L1 L2L2 The Limit of a Function If the function values as x approaches a from each side of a do not yield the same function value, the function does not exist.

|a|a |x|x |x|x   The Limit of a Function If as x approaches a from either side of a, f (x) goes to either infinity or negative infinity, the limit as x approaches a of f (x) is positive or negative infinity respectively. f → ∞

|x|x The Limit of a Function If as x approaches infinity (or negative infinity), f (x) approaches L, then the limit as x approaches a of f (x) is L.  L f (x 1 ) x → ∞

|a|a |b|b |a|a |b|b o Continuity A function is continuous over an interval of x values if it has no breaks, gaps, nor vertical asymptotes on that interval. Continuous on (a, b) |c|c Not Continuous on (a, b) since discontinuous at x = c

|a|a |b|b |a|a |b|b o Continuity In other words, a function is continuous at x = c if the following is true. Continuous on (a, b) |c|c Not Continuous on (a, b) since discontinuous at x = c

|c|c |a|a |b|b Continuity Types Is the following continuous? Infinite Discontinuity |c|c |a|a |b|b Jump Discontinuity

|a|a |b|b o Continuity Types Is the function continuous? |c|c Removable Discontinuity