Section 1.3 Limits Done RightExample: Give the largest δ that works with ε = 0.1 for the limit ()lim→ −− =x 11 2x 3Example: Give the largest δ that works with ε = 0.1 for the limit ()lim→ −+ = −x 14 x 2 2Techniques for evaluating Limits 1. Direct substitution – If you can – plug it in, plug it in. ()lim→+ =2x 33x 1 ()lim→+ =2x 3x x lim→ +=  + 2x 03x 2x2x 12. Functions that agree at all but one point a. Cancellation technique (factor and cancel) lim→ − − −=  + 2x 12x x 3x 1 −3 −2 −1 1 2−6−5−4−3−2−11lim→ −= − 3x 2x 2x 8 lim→ − +=  − 22x 1x xx 1b. Rationalization technique (rationalize the numerator). lim→+ −−x 3x 1 2x 3 −2 −1 1 2 3 4 51lim→+ −−2x 1x 1 2x 1 −3 −2 −1 1 2 3−3−2−1123Limits that Fail to Exist 1. Behavior differs from the left and right. 2. Unbounded behavior. 3. Oscillating behavior.Left-handed and Right-handed limits • Formal definitions similar to formal definition of a limit (see page 6 in 1.3) • Notation: limx→c+f (x)limx→c−f (x)Uniqueness of a Limit If ()lim f→=x cx L and ()lim f→=x cx M then L = M.The Limit of the Sum….. If ()lim f→=x cx L and ()lim g→=x cx M then ()()()lim f g→+ = +x cx x L M is the sum of the limits (provided the limits exist).The Limit of the Difference….. If ()lim f→=x cx L and ()lim g→=x cx M then ()()()lim f g→− = −x cx x L M is the difference of the limits (provided the limits exist).The Limit of the Product….. If ()lim f→=x cx L and ()lim g→=x cx M then ()()()lim f g→=x cx x LM is the product of the limits (provided the limits exist).The Limit of the Quotients….. If ()lim f→=x cx L and ()lim g→=x cx M and M ≠ 0 then ()( )flimg→=x cxLx M is the quotient of the limits (provided the limits exist).More on Quotients…. If ()lim f→=x cx L and ()lim g→=x cx M and L ≠ 0 and M = 0, then ()( )flimg→x cxdoes not existxGraph f (x) = 3. ()lim f lim→ →= =x 5 x 5x 3 What about x = 18? What about any real value for x? lim→=x ca a, where a and c are constants.Graph f (x) = x. What is ()lim f lim→ →= =x 5 x 5x x What about x = 18? What about any real value for x? lim→=x cx c, where c is a constant.Some more: If ( ),f,−≠=−=4t 8t 2tt 27 t 2 then ()lim f→=t 2tlim→−+x 01 1x 4 4xLimits as x→∞ limx→∞1x= limx→∞1x2= limx→∞1xn=limx→∞2x2− 3x +14x − x2=Section 1.4 Continuity----- what is it?A function f is said to be continuous at a point c if 1) f ( c ) is defined. 2) ()lim f→x cxexists (limit from the left equals the limit from the right) 3) f ( c ) = limx→ cf x() This is the three step method to prove continuity. c1) f ( c ) is defined 2) ()lim f x→x c exists 3) f ( c ) = ()lim f x→x c c c c cTypes of Discontinuity Removable and Non-Removable (Jump, Infinite) Which is which? f (c) f (x) x y L c f (x) x y cTheorem If f and g are continuous at c, then i) f + g is continuous at c ii) f – g is continuous at c iii) αf is continuous at c for each real α (a number) iv) f⋅g is continuous at c v) f /g is continuous at c provided ()c 0g≠ Theorem - If g is continuous at c and f is continuous at g(c), then the composition is continuous at c.Definition: One-Sided Continuity A function f is called continuous from the left at c if limx→c−f x()=f c(). It is called continuous from the right at c if limx→c+f x()=f c(). f is continuous at c iff f (c), ()lim f−→x cx, and ()lim f+→x cx all exist and are equal.Where are polynomials continuous? Where are rational functions continuous? Discuss continuity for: f x( )=x+2x2−x−6f t( )=t2+ 2tt2−4f x( )=x+5x2+5 f x( )=x+5x2+5x f x()= xf x( )=x −1x2+4x−5Use the 3 step method to determine continuity or discontinuity. f x( )=x3x < 1x x ≥ 1h x( )=1 x ≤ −2−1 x > −2p x( )=4 − x2− 2 ≤ x < 23 x = 2x − 2 x > 2Determine if the following function is continuous at the point where x = 3.Find c so that R(x) is continuous. R x( )=2x - 3 x < 2cx - x2x ≥ 2Find A and B so that f (x) is continuous. ( )f− < −= = −− > −22 x 1 x 2x A x 2Bx 3 x 2Discuss the continuity of 2213 1( )2 1 11/ 1x xxg xx xx x− < −= −=− − <