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

lim

𝑥→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

lim

𝑥→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=𝜋2 𝑥

h𝑇 𝑒𝑟𝑒𝑖𝑠𝑛𝑜𝑟𝑒𝑚𝑜𝑣𝑎𝑏𝑙𝑒 𝑑𝑖𝑠𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑖𝑡𝑦 𝑓𝑜𝑟 𝑓 (𝑥)

𝑓 (𝑥 )=𝑡𝑎𝑛( 𝜋2 𝑥 )

𝑐𝑜𝑠𝜃=0

𝜃=𝜋2 ,3𝜋2 , 5𝜋2 ,⋯

𝜋2 𝑥=

𝜋2 ,3𝜋2 , 5𝜋2 ,⋯

𝑥=1 ,3 ,5 ,7 ,⋯

𝑓 (𝑥 )=𝑠𝑖𝑛( 𝜋2 𝑥)𝑐𝑜𝑠 (𝜋2 𝑥)

2.5 – Continuity

ExamplesRemovable 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=5lim𝑥→∞

𝑓 (𝑥 )=−2

𝐻 . 𝐴 .𝑎𝑡 𝑦=−2

lim𝑥→∞

𝑓 (𝑥 )=0

𝐻 . 𝐴 .𝑎𝑡 𝑦=0lim

𝑥→ 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

ExamplesLimits Involving Infinity

2.6 – Limits Involving Infinity; Asymptotes of GraphsOblique Asymptotes