Do Now – Graph:

One-Sided Limits, Sandwich Theorem Section 2.1b

One-Sided and Two-Sided Limits Sometimes the values of a function tend to different limits as x approaches a number c from opposite sides… Right-hand Limit – the limit of a function f as x approaches c from the right. + Left-hand Limit – the limit of a function f as x approaches c from the left. –

One-Sided and Two-Sided Limits We sometimes call the two-sided limits of f at c to distinguish it from the one-sided limits from the right and left. Theorem A function f(x) has a limit as x approaches c if and only if the right-hand and left-hand limits at c exist and are equal. In symbols: + and –

The Sandwich Theorem If for all in some interval about c, and then If we cannot find a limit directly, we may be able to use this theorem to find it indirectly… If for all in some interval about c, and then

The Sandwich Theorem y h f L g x c Graphically………Sandwiching f between g and h forces the limiting value of f to be between the limiting values of g and h: y h f L g x c

Guided Practice First, sketch a graph of the greatest integer function, then find each of the given limits. 2 1 –2 –1 1 2 3 –1 –2

Guided Practice Sketch a graph of the given function, then evaluate limits for the function at x = 0, 1, 2, 3, and 4. Note: If f is not defined to the left of x = c, then f does not have a left-hand limit at c. Similarly, if f is not defined to the right of x = c, then f does not have a right-hand limit at c.

Guided Practice Sketch a graph of the given function, then evaluate limits for the function at x = 0, 1, 2, 3, and 4. At x = 0

Guided Practice At x = 1 even though f(1) = 1 Sketch a graph of the given function, then evaluate limits for the function at x = 0, 1, 2, 3, and 4. At x = 1 even though f(1) = 1 Note: f has no limit as x  1 (why not???)

Guided Practice At x = 2 Note: even though f(2) = 2 Sketch a graph of the given function, then evaluate limits for the function at x = 0, 1, 2, 3, and 4. At x = 2 Note: even though f(2) = 2

Guided Practice Sketch a graph of the given function, then evaluate limits for the function at x = 0, 1, 2, 3, and 4. At x = 3

Guided Practice Sketch a graph of the given function, then evaluate limits for the function at x = 0, 1, 2, 3, and 4. At x = 4 Note: At non-integer values of c between 0 and 4, the function has a limit as x  c.

Guided Practice For the following, (a) draw the graph of f, (b) determine the left- and right-hand limits at c, and (c) determine if the limit as x approaches c exists. Explain your reasoning.

Guided Practice For the following, (a) draw the graph of f, (b) determine the left- and right-hand limits at c, and (c) determine if the limit as x approaches c exists. Explain your reasoning.

Guided Practice For the following, draw the graph of f, and answer: (a) At what points c in the domain of f does lim x  c exist? (b) At what points c does only the left-hand limit exist? (c) At what points c does only the right-hand limit exist? (0,1) (a) (b) (c)