1. 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

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

3. 2.5 – Continuity

4. 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. ∴

5. 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. ∴

6. 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. ∴

7. 2.5 – Continuity    𝑥=0 (left end point) 𝑓 𝑐 𝑒𝑥𝑖𝑠𝑡𝑠 𝑓 0 =1
lim 𝑥→ 𝑐 + 𝑓 𝑥 𝑒𝑥𝑖𝑠𝑡𝑠
lim 𝑥→ 𝑓 𝑥 =1 
lim 𝑥→ 𝑐 + 𝑓 𝑥 =𝑓(𝑐)
1=1 ∴
𝑓 𝑥 𝑖𝑠 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑎𝑡 𝑡ℎ𝑒 𝑙𝑒𝑓𝑡 𝑒𝑛𝑑𝑝𝑜𝑖𝑛𝑡, 𝑥=0.

8. 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.

9. 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.

10. 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.

11. 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.

12. 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 , ⋯

13. Removable Discontinuity
Examples

14. 2.5 – Continuity 𝑓 𝑥 =2 𝑥 3 −16 𝑥 2 +38𝑥−22 1,5
𝑓 𝑥 =2 𝑥 3 −16 𝑥 2 +38𝑥− ,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

15. 2.6 – Limits Involving Infinity; Asymptotes of Graphs
𝐴𝑠 𝑥→ 0 + , 𝑦→∞
lim 𝑥→ 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 =−∞

16. 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→ 𝑓 𝑥 =∞
lim 𝑥→ −7 + 𝑓 𝑥 =∞
𝐻.𝐴. 𝑎𝑡 𝑦=−2
𝑉.𝐴. 𝑎𝑡 𝑥=2
𝑉.𝐴. 𝑎𝑡 𝑥=−7

17. 2.6 – Limits Involving Infinity; Asymptotes of Graphs

18. 2.6 – Limits Involving Infinity; Asymptotes of Graphs
Examples

19. 2.6 – Limits Involving Infinity; Asymptotes of Graphs
Oblique Asymptotes