2.5 – Continuity A continuous function is one that can be plotted without the plot being broken. Is the graph of f(x) a continuous function on the interval [0, 4]? No At what values of x is the function discontinuous and why? 𝑥=1 𝑇ℎ𝑒𝑟𝑒 𝑖𝑠 𝑎 𝑗𝑢𝑚𝑝. 𝑥=2 𝑇ℎ𝑒𝑟𝑒 𝑖𝑠 𝑎 ℎ𝑜𝑙𝑒. 𝑥=4 𝑇ℎ𝑒𝑟𝑒 𝑖𝑠 𝑎 𝑗𝑢𝑚𝑝. Is the graph of f(x) continuous at 𝑥=3? Yes

What are the rules for continuity at a point? lim 𝑥→ 3 − 𝑓 𝑥 = lim 𝑥→ 3 + 𝑓 𝑥 = 2 2 lim 𝑥→3 𝑓 𝑥 = 𝑓 3 = 2 2 lim 𝑥→ 1 − 𝑓 𝑥 = lim 𝑥→ 1 + 𝑓 𝑥 = 1 lim 𝑥→1 𝑓 𝑥 = 𝑓 1 = 𝐷𝑁𝐸 1 lim 𝑥→ 2 − 𝑓 𝑥 = 1 lim 𝑥→ 2 + 𝑓 𝑥 = 1 lim 𝑥→2 𝑓 𝑥 = 1 𝑓 2 = 2 What are the rules for continuity at a point? lim 𝑥→ 4 − 𝑓 𝑥 = 1 lim 𝑥→ 4 + 𝑓 𝑥 = 𝑛𝑜𝑛𝑒 lim 𝑥→4 𝑓 𝑥 = 𝑓 4 = 𝑛𝑜𝑛𝑒 0.5

2.5 – Continuity

2.5 – Continuity   𝑥=1 𝑓 1 =1 𝑓 𝑐 𝑒𝑥𝑖𝑠𝑡𝑠 lim 𝑥→𝑐 𝑓 𝑥 𝑒𝑥𝑖𝑠𝑡𝑠 𝑓 𝑥 𝑖𝑠 𝑛𝑜𝑡 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑎𝑡 𝑥=1. ∴

2.5 – Continuity    𝑥=2 𝑓 𝑐 𝑒𝑥𝑖𝑠𝑡𝑠 𝑓 2 =2 lim 𝑥→𝑐 𝑓 𝑥 𝑒𝑥𝑖𝑠𝑡𝑠 2≠1 𝑓 𝑥 𝑖𝑠 𝑛𝑜𝑡 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑎𝑡 𝑥=2. ∴

2.5 – Continuity    𝑥=3 𝑓 𝑐 𝑒𝑥𝑖𝑠𝑡𝑠 𝑓 2 =2 lim 𝑥→𝑐 𝑓 𝑥 𝑒𝑥𝑖𝑠𝑡𝑠 2=2 𝑓 𝑥 𝑖𝑠 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑎𝑡 𝑥=3. ∴

2.5 – Continuity    𝑥=0 (left end point) 𝑓 𝑐 𝑒𝑥𝑖𝑠𝑡𝑠 𝑓 0 =1 lim 𝑥→ 𝑐 + 𝑓 𝑥 𝑒𝑥𝑖𝑠𝑡𝑠 lim 𝑥→ 0 + 𝑓 𝑥 =1  lim 𝑥→ 𝑐 + 𝑓 𝑥 =𝑓(𝑐) 1=1 ∴ 𝑓 𝑥 𝑖𝑠 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑎𝑡 𝑡ℎ𝑒 𝑙𝑒𝑓𝑡 𝑒𝑛𝑑𝑝𝑜𝑖𝑛𝑡, 𝑥=0.

2.5 – Continuity    𝑥=4 (right end point) 𝑓 𝑐 𝑒𝑥𝑖𝑠𝑡𝑠 𝑓 4 =0.5 lim 𝑥→ 𝑐 − 𝑓 𝑥 𝑒𝑥𝑖𝑠𝑡𝑠 lim 𝑥→ 4 − 𝑓 𝑥 =1  lim 𝑥→ 𝑐 + 𝑓 𝑥 =𝑓(𝑐) 0.5≠1 ∴ 𝑓 𝑥 𝑖𝑠 𝑛𝑜𝑡 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑎𝑡 𝑡ℎ𝑒 𝑟𝑖𝑔ℎ𝑡 𝑒𝑛𝑑𝑝𝑜𝑖𝑛𝑡, 𝑥=4.

Removable Discontinuity Removable discontinuity occurs at a point where the function has a hole but does not have a function value. 𝐶𝑟𝑒𝑎𝑡𝑒 𝑎 𝑠𝑖𝑚𝑖𝑙𝑎𝑟 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛, 𝑔 𝑥 , 𝑡ℎ𝑎𝑡 𝑖𝑠 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑎𝑡 𝑥=2. lim 𝑥→2 𝑔 𝑥 𝑒𝑥𝑖𝑠𝑡𝑠 lim 𝑥→2 𝑔 𝑥 =1 lim 𝑥→2 𝑔 𝑥 =𝑔(2) 1=1  𝑔 2 𝑒𝑥𝑖𝑠𝑡𝑠 𝑔 2 =1 𝑔(𝑥) 𝑖𝑓 𝑥≠2 1 𝑖𝑓 𝑥=2 𝑔 𝑥 = 𝑓 𝑥 𝑖𝑠 𝑑𝑖𝑠𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑎𝑡 𝑥=2 𝐴 ℎ𝑜𝑙𝑒 𝑒𝑥𝑖𝑠𝑡 𝑎𝑡 𝑥=2. 𝑔 𝑥 𝑖𝑠 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑎𝑡 𝑥=2. 𝑅𝑒𝑚𝑜𝑣𝑎𝑏𝑙𝑒 𝑑𝑖𝑠𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑖𝑡𝑦 𝑎𝑡 𝑥=2.

Removable Discontinuity Removable discontinuity occurs at a point where the function has a hole but does not have a function value. 𝐶𝑟𝑒𝑎𝑡𝑒 𝑎 𝑠𝑖𝑚𝑖𝑙𝑎𝑟 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛, 𝑔 𝑥 , 𝑡ℎ𝑎𝑡 𝑖𝑠 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑎𝑡 𝑥=3. 𝑓(𝑥) lim 𝑥→3 𝑔 𝑥 𝑒𝑥𝑖𝑠𝑡𝑠 lim 𝑥→3 𝑔 𝑥 =0  lim 𝑥→3 𝑔 𝑥 =𝑔(3) 0=0 𝑔 3 𝑒𝑥𝑖𝑠𝑡𝑠 𝑔 3 =0 𝑓 𝑥 𝑖𝑠 𝑑𝑖𝑠𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑎𝑡 𝑥=3 𝑔(𝑥) 𝑖𝑓 𝑥≠3 0 𝑖𝑓 𝑥=3 𝑔 𝑥 = 𝐴 ℎ𝑜𝑙𝑒 𝑒𝑥𝑖𝑠𝑡 𝑎𝑡 𝑥=3. 𝑔 𝑥 𝑖𝑠 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑎𝑡 𝑥=3. 𝑅𝑒𝑚𝑜𝑣𝑎𝑏𝑙𝑒 𝑑𝑖𝑠𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑖𝑡𝑦 𝑎𝑡 𝑥=3.

Removable Discontinuity Example: The given function is discontinuous. Where is it discontinuous and is it removable? 𝑓 𝑥 = 𝑥 2 −4 𝑥−2 𝑥−2=0 𝑥=2 𝑓(𝑥) 𝑖𝑠 𝑑𝑖𝑠𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑎𝑡 𝑥=2 𝑓 𝑥 = 𝑥−2 𝑥+2 𝑥−2 𝑇ℎ𝑒 𝑓𝑎𝑐𝑡𝑜𝑟 𝑥−2 𝑐𝑎𝑛𝑐𝑒𝑙𝑠. 𝑇ℎ𝑒𝑟𝑒 𝑖𝑠 𝑎 ℎ𝑜𝑙𝑒 𝑎𝑡 𝑥=2. 𝐼𝑡 𝑖𝑠 𝑟𝑒𝑚𝑜𝑣𝑎𝑏𝑙𝑒 𝑎𝑡 𝑥=2.

Removable Discontinuity Example: The given function is discontinuous. Where is it discontinuous and is it removable? 𝑓 𝑥 =𝑡𝑎𝑛 𝜋 2 𝑥 𝜋 2 𝑥= 𝜋 2 , 3𝜋 2 , 5𝜋 2 , ⋯ 𝑥=1, 3, 5, 7, ⋯ 𝑓 𝑥 = 𝑠𝑖𝑛 𝜋 2 𝑥 𝑐𝑜𝑠 𝜋 2 𝑥 𝑓 𝑥 𝑖𝑠 𝑑𝑖𝑠𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑎𝑡 𝑥=1,3, 5, 7, ⋯ 𝑓 𝑥 = 𝑠𝑖𝑛 𝜋 2 𝑥 𝑐𝑜𝑠 𝜋 2 𝑥 𝑐𝑜𝑠 𝜋 2 𝑥 =0 𝑇ℎ𝑒 𝑓𝑎𝑐𝑡𝑜𝑟 𝑐𝑜𝑠 𝜋 2 𝑥 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑐𝑎𝑛𝑐𝑒𝑙. 𝑙𝑒𝑡 𝜃= 𝜋 2 𝑥 𝑐𝑜𝑠𝜃=0 𝑇ℎ𝑒𝑟𝑒 𝑖𝑠 𝑛𝑜 𝑟𝑒𝑚𝑜𝑣𝑎𝑏𝑙𝑒 𝑑𝑖𝑠𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑖𝑡𝑦 𝑓𝑜𝑟 𝑓(𝑥) 𝜃= 𝜋 2 , 3𝜋 2 , 5𝜋 2 , ⋯

Removable Discontinuity Examples

2.5 – Continuity 𝑓 𝑥 =2 𝑥 3 −16 𝑥 2 +38𝑥−22 1,5 𝑓 𝑥 =2 𝑥 3 −16 𝑥 2 +38𝑥−22 1,5 𝑓 𝑥 𝑖𝑠 𝑎 𝑝𝑜𝑙𝑦𝑛𝑜𝑛𝑚𝑖𝑎𝑙 ∴𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑓 1 = 2 𝑓 5 = 18 𝑓 𝑥 =8 8=2 𝑥 3 −16 𝑥 2 +38𝑥−22 𝑥=4.547

2.6 – Limits Involving Infinity; Asymptotes of Graphs 𝐴𝑠 𝑥→ 0 + , 𝑦→∞ lim 𝑥→ 0 + 𝑓 𝑥 =∞ 𝐴𝑠 𝑥→−∞, 𝑦→0 𝐴𝑠 𝑥→∞, 𝑦→0 lim 𝑥→−∞ 𝑓 𝑥 =0 lim 𝑥→∞ 𝑓 𝑥 =0 𝐴𝑠 𝑥→ 0 − , 𝑦→−∞ lim 𝑥→ 0 − 𝑓 𝑥 =−∞

2.6 – Limits Involving Infinity; Asymptotes of Graphs 𝐻.𝐴. 𝑎𝑡 𝑦=5 𝐻.𝐴. 𝑎𝑡 𝑦=0 𝑉.𝐴. 𝑎𝑡 𝑥=4 lim 𝑥→∞ 𝑓 𝑥 =−2 lim 𝑥→ 2 + 𝑓 𝑥 =∞ lim 𝑥→ −7 + 𝑓 𝑥 =∞ 𝐻.𝐴. 𝑎𝑡 𝑦=−2 𝑉.𝐴. 𝑎𝑡 𝑥=2 𝑉.𝐴. 𝑎𝑡 𝑥=−7

2.6 – Limits Involving Infinity; Asymptotes of Graphs

2.6 – Limits Involving Infinity; Asymptotes of Graphs Examples

2.6 – Limits Involving Infinity; Asymptotes of Graphs Oblique Asymptotes