Limits of Functions and Continuity

1. Limits of Functions and Continuity

2. 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 (x1) f (x1) o f(a) ≠ L f (a) = L   f (x2) f (x2) | x1 | a | x2 | x1 | a | x2

3. 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.  L2  L1 | x | a | x

4. 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 | a | x

5. 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. f (x1)  L | x x→ ∞

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

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

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

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