2.5 – Continuity A continuous function is one that can be plotted without the plot being broken....
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Transcript of 2.5 – Continuity A continuous function is one that can be plotted without the plot being broken....
2.5 – ContinuityA 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 h𝑇 𝑒𝑟𝑒𝑖𝑠 𝑎 𝑗𝑢𝑚𝑝 .𝑥=2 h𝑇 𝑒𝑟𝑒𝑖𝑠 𝑎h𝑜𝑙𝑒 .𝑥=4 h𝑇 𝑒𝑟𝑒𝑖𝑠 𝑎 𝑗𝑢𝑚𝑝 .
Is the graph of f(x) continuous at ? Yes
2.5 – Continuity
lim𝑥→ 3−
𝑓 (𝑥 )=¿¿2 lim𝑥→3+¿ 𝑓 (𝑥 )=¿¿ ¿
¿2
lim𝑥→ 3
𝑓 (𝑥 )=¿¿2 𝑓 (3 )=¿2
lim𝑥→ 1−
𝑓 (𝑥 )=¿ ¿0 lim𝑥→1+¿ 𝑓 (𝑥 )=¿ ¿¿
¿1
lim𝑥→ 1
𝑓 (𝑥 )=¿¿𝐷𝑁𝐸 𝑓 (1 )=¿1
lim𝑥→ 2−
𝑓 (𝑥 )=¿¿1 lim𝑥→2+¿ 𝑓 (𝑥 )=¿ ¿ ¿
¿1
lim𝑥→ 2
𝑓 (𝑥 )=¿¿1 𝑓 (2 )=¿2
lim𝑥→ 4−
𝑓 (𝑥 )=¿¿1 lim𝑥→4+¿ 𝑓 (𝑥 )=¿ ¿¿
¿𝑛𝑜𝑛𝑒
lim𝑥→ 4
𝑓 (𝑥 )=¿¿𝑛𝑜𝑛𝑒 𝑓 (4 )=¿0.5
What are the rules for continuity at a point?
2.5 – Continuity
2.5 – Continuity
lim𝑥→𝑐
𝑓 (𝑥 )𝑒𝑥𝑖𝑠𝑡𝑠
∴ 𝑓 (𝑥 ) 𝑖𝑠𝑛𝑜𝑡 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠𝑎𝑡 𝑥=1.
lim𝑥→ 1
𝑓 (𝑥 )=𝐷𝑁𝐸
𝑓 (1 )=1𝑥=1𝑓 (𝑐 )𝑒𝑥𝑖𝑠𝑡𝑠
2.5 – Continuity
lim𝑥→𝑐
𝑓 (𝑥 )𝑒𝑥𝑖𝑠𝑡𝑠
∴ 𝑓 (𝑥 ) 𝑖𝑠𝑛𝑜𝑡 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠𝑎𝑡 𝑥=2.
lim𝑥→ 2
𝑓 (𝑥 )=1
𝑓 (2 )=2𝑥=2𝑓 (𝑐 )𝑒𝑥𝑖𝑠𝑡𝑠
lim𝑥→𝑐
𝑓 (𝑥 )= 𝑓 (𝑐) 2≠1
2.5 – Continuity
lim𝑥→𝑐
𝑓 (𝑥 )𝑒𝑥𝑖𝑠𝑡𝑠
∴ 𝑓 (𝑥 ) 𝑖𝑠𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠𝑎𝑡 𝑥=3.
lim𝑥→ 3
𝑓 (𝑥 )=2
𝑓 (2 )=2𝑥=3𝑓 (𝑐 )𝑒𝑥𝑖𝑠𝑡𝑠
lim𝑥→𝑐
𝑓 (𝑥 )= 𝑓 (𝑐) 2=2
2.5 – Continuity
lim𝑥→𝑐+¿ 𝑓 (𝑥 )𝑒𝑥𝑖𝑠𝑡𝑠¿
¿
∴ 𝑓 (𝑥 ) 𝑖𝑠𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠𝑎𝑡 h𝑡 𝑒𝑙𝑒𝑓𝑡 𝑒𝑛𝑑𝑝𝑜𝑖𝑛𝑡 ,𝑥=0.
lim𝑥→0+¿ 𝑓 (𝑥 )=1¿
¿
𝑓 (0 )=1𝑥=0 (¿ point )𝑓 (𝑐 )𝑒𝑥𝑖𝑠𝑡𝑠
lim𝑥→𝑐+¿ 𝑓 (𝑥 )= 𝑓 (𝑐 )¿
¿ 1=1
2.5 – Continuity
lim𝑥→𝑐−
𝑓 (𝑥 )𝑒𝑥𝑖𝑠𝑡𝑠
∴ 𝑓 (𝑥 ) 𝑖𝑠𝑛𝑜𝑡 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠𝑎𝑡 h𝑡 𝑒 h𝑟𝑖𝑔 𝑡 𝑒𝑛𝑑𝑝𝑜𝑖𝑛𝑡 ,𝑥=4.
lim𝑥→ 4−
𝑓 (𝑥 )=1
𝑓 (4 )=0.5𝑥=4(¿end point )𝑓 (𝑐 )𝑒𝑥𝑖𝑠𝑡𝑠
lim𝑥→𝑐+¿ 𝑓 (𝑥 )= 𝑓 (𝑐 )¿
¿ 0.5≠1
2.5 – ContinuityRemovable Discontinuity
Removable discontinuity occurs at a point where the function has a hole but does not have a function value.
𝑔 (𝑥 )𝑖𝑠 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠𝑎𝑡 𝑥=2.𝐴 h𝑜𝑙𝑒𝑒𝑥𝑖𝑠𝑡 𝑎𝑡 𝑥=2.𝑓 (𝑥 ) 𝑖𝑠 𝑑𝑖𝑠𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠𝑎𝑡 𝑥=2
𝑔 (𝑥 )=¿{𝑔 (𝑥) 𝑖𝑓 𝑥≠21 𝑖𝑓 𝑥=2
𝑅𝑒𝑚𝑜𝑣𝑎𝑏𝑙𝑒𝑑𝑖𝑠𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑖𝑡𝑦 𝑎𝑡 𝑥=2.
lim𝑥→ 2
𝑔 (𝑥 )𝑒𝑥𝑖𝑠𝑡𝑠 lim𝑥→ 2
𝑔 (𝑥 )=1
𝑔 (2 )=1𝑔 (2 )𝑒𝑥𝑖𝑠𝑡𝑠
lim𝑥→ 2
𝑔 (𝑥 )=𝑔 (2) 1=1
2.5 – ContinuityRemovable Discontinuity
Removable discontinuity occurs at a point where the function has a hole but does not have a function value.
𝑔 (𝑥 )𝑖𝑠 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠𝑎𝑡 𝑥=3.𝐴 h𝑜𝑙𝑒𝑒𝑥𝑖𝑠𝑡 𝑎𝑡 𝑥=3.𝑓 (𝑥 ) 𝑖𝑠 𝑑𝑖𝑠𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠𝑎𝑡 𝑥=3 𝑔 (𝑥 )=¿{𝑔 (𝑥) 𝑖𝑓 𝑥≠3
0 𝑖𝑓 𝑥=3
𝑅𝑒𝑚𝑜𝑣𝑎𝑏𝑙𝑒𝑑𝑖𝑠𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑖𝑡𝑦 𝑎𝑡 𝑥=3.
lim𝑥→ 3
𝑔 (𝑥 )𝑒𝑥𝑖𝑠𝑡𝑠 lim𝑥→ 3
𝑔 (𝑥 )=0
𝑔 (3 )=0𝑔 (3 )𝑒𝑥𝑖𝑠𝑡𝑠
lim𝑥→ 3
𝑔 (𝑥 )=𝑔(3) 0=0
𝑓 (𝑥)
2.5 – Continuity
Example:
𝑓 (𝑥 )=𝑥2−4𝑥−2
𝑓 (𝑥) 𝑖𝑠 𝑑𝑖𝑠𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠𝑎𝑡 𝑥=2
𝑓 (𝑥 )= (𝑥−2 ) (𝑥+2 )𝑥−2
h𝑇 𝑒 𝑓𝑎𝑐𝑡𝑜𝑟 (𝑥−2 )𝑐𝑎𝑛𝑐𝑒𝑙𝑠 .
Removable Discontinuity
The given function is discontinuous. Where is it discontinuous and is it removable?
𝑥−2=0𝑥=2
h𝑇 𝑒𝑟𝑒𝑖𝑠 𝑎h𝑜𝑙𝑒𝑎𝑡 𝑥=2 ..
2.5 – Continuity
Example:
𝑓 (𝑥 )=𝑠𝑖𝑛( 𝜋2 𝑥)𝑐𝑜𝑠 (𝜋2 𝑥) 𝑓 (𝑥 ) 𝑖𝑠 𝑑𝑖𝑠𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠𝑎𝑡 𝑥=1,3 ,5 ,7 ,⋯
h𝑇 𝑒 𝑓𝑎𝑐𝑡𝑜𝑟 (𝑐𝑜𝑠( 𝜋2 𝑥))𝑑𝑜𝑒𝑠𝑛𝑜𝑡 𝑐𝑎𝑛𝑐𝑒𝑙 .
Removable Discontinuity
The given function is discontinuous. Where is it discontinuous and is it removable?
𝑐𝑜𝑠( 𝜋2 𝑥)=0
𝑙𝑒𝑡 𝜃=𝜋2𝑥
h𝑇 𝑒𝑟𝑒𝑖𝑠𝑛𝑜𝑟𝑒𝑚𝑜𝑣𝑎𝑏𝑙𝑒 𝑑𝑖𝑠𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑖𝑡𝑦 𝑓𝑜𝑟 𝑓 (𝑥)
𝑓 (𝑥 )=𝑡𝑎𝑛( 𝜋2 𝑥 )
𝑐𝑜𝑠𝜃=0
𝜃=𝜋2,3𝜋2,5𝜋2,⋯
𝜋2𝑥=
𝜋2,3𝜋2,5𝜋2,⋯
𝑥=1 ,3 ,5 ,7 ,⋯
𝑓 (𝑥 )=𝑠𝑖𝑛( 𝜋2 𝑥)𝑐𝑜𝑠 (𝜋2 𝑥)
2.5 – Continuity
Examples
Removable Discontinuity
2.5 – Continuity
𝑓 (𝑥 )=2𝑥3−16 𝑥2+38𝑥−22 [1,5 ]𝑓 (1 )=¿2 𝑓 (5 )=¿18 𝑓 (𝑥 )=8
8=2 𝑥3−16 𝑥2+38 𝑥−22 𝑥=4.547
𝑓 (𝑥 ) 𝑖𝑠𝑎𝑝𝑜𝑙𝑦𝑛𝑜𝑛𝑚𝑖𝑎𝑙∴𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠
2.6 – Limits Involving Infinity; Asymptotes of Graphs
𝐴𝑠𝑥→∞, 𝑦→0𝐴𝑠𝑥→−∞ , 𝑦→0
𝐴𝑠𝑥→0− , 𝑦→−∞
𝐴𝑠𝑥→0+¿ , 𝑦→∞¿
lim𝑥→0+¿ 𝑓 (𝑥 )=∞¿
¿
lim𝑥→ 0−
𝑓 (𝑥 )=−∞
lim𝑥→∞
𝑓 (𝑥 )=0lim𝑥→−∞
𝑓 (𝑥 )=0
2.6 – Limits Involving Infinity; Asymptotes of Graphs
lim𝑥→−∞
𝑓 (𝑥 )=5
𝐻 . 𝐴 .𝑎𝑡 𝑦=5
lim𝑥→∞
𝑓 (𝑥 )=−2
𝐻 . 𝐴 .𝑎𝑡 𝑦=−2
lim𝑥→∞
𝑓 (𝑥 )=0
𝐻 . 𝐴 .𝑎𝑡 𝑦=0
lim𝑥→2+¿ 𝑓 (𝑥 )=∞¿
¿
𝑉 .𝐴 .𝑎𝑡 𝑥=2
lim𝑥→ 4−
𝑓 (𝑥 )=−∞
𝑉 .𝐴 .𝑎𝑡 𝑥=4lim
𝑥→−7+¿ 𝑓 (𝑥 )=∞¿¿
𝑉 .𝐴 .𝑎𝑡 𝑥=−7
2.6 – Limits Involving Infinity; Asymptotes of Graphs
2.6 – Limits Involving Infinity; Asymptotes of Graphs
Examples
Limits Involving Infinity
2.6 – Limits Involving Infinity; Asymptotes of Graphs
Oblique Asymptotes