Continuity(worksheet(
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Intuitively,!a!function!is!not!continuous!anywhere!it!“jumps”.!!
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More!precisely,!a!function!f!is!continuous!at!a!point!x#=#C!if!(and!only!if)
lim− f ( x ) = lim+ f ( x ) = f ( C ) .!
x→C
x→C
!
The!function!f#pictured!appears!to!be!discontinuous!at!x#=#0,!2,!4!and!6.!!
Verify!this!result!using!the!definition!above!and!those!4!values!for!C.!!
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1) lim− f ( x ) = !
x→6
lim f ( x ) = !
x→6 +
f (6) = !
!
!
!
2) lim− f ( x ) = !
x→4
lim f ( x ) = !
x→4 +
f (4) = !
!
!
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3) ! lim− f ( x ) = !
x→2
lim f ( x ) = !
x→2 +
f (2) = !
!
!
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4) lim− f ( x ) = !
x→0
lim f ( x ) = !
x→0 +
f (0) = !
!
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5) Show!that!f#is!continuous!at!x#=#&1!by!evaluating!the!following:!
lim− f ( x ) = !
x→−1
lim f ( x ) = !
x→−1+
f ( −1) = !
!
!
So!we!can!use!limits!to!determine!if!a!function!is!continuous!at!a!point.!However,!it!is!far!more!
useful!to!use!continuity!to!determine!limits!in!the!first!place.!!
x2 − 4
,!finding! lim g(x) !is!(relatively)!hard.!In!2.2,!our!only!plan!of!attack!was!to!guess!
x→2
x−2
the!limit!by!evaluating!something!like! g(2.000001) !and! g(1.99999) and!taking!a!guess.!!
If!! g(x) =
However,!if!we!notice!that! lim g(x) = lim h(x) !for! h(x) = x + 2 ,!and!that!h!is!continuous!at!x#=#2,!we!
x→2
x→2
can!use!continuity!to!evaluate!these!limits!as!simply! h ( 2 ) .!!
This!is!because!the!fact!that!h!is!continuous!at!x#=!2!tells!us!(by!definition)!that!
lim− h ( x ) = lim+ h ( x ) = h ( 2 ) ,!so! h ( 2 ) = lim h ( x ) = lim g ( x ) !
x→2
x→2
x→2
x→2
Going!forward,!we!will!often!be!faced!with!evaluating!limits!at!points!of!discontinuity.!We!will!
typically!evaluate!functions!(like!g!above)!at!a!point!of!discontinuity!(like!x#=#2)!by!finding!a!
function!(like!h#above)!that!has!the!same!limits,!but!is!continuous!at!the!point!in!question.!!
Use!these!methods!to!evaluate!the!following:!
2x 2 − 5x − 3
1)! lim
!! !
x→3
x−3
!
!
!
!
x 3 + 5x 2 + 10x + 8
2)! lim
!
x→−2
x+2
!
1
1
−
3)! lim 2 2 + x !
x→0
x
!
!
5)! lim
!
!
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4)! lim
x→4
x −2
!!
x−4
!
!
!
!
x→5
x −1 − 2
!
x−5
Hint:!Multiply!the!top!and!the!bottom!by!the!conjugate!of!the!top.!
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6)! lim
x→10
x − 10
!!
x − 10
!
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7)! lim
Hint:!Evaluate!the!two!oneTsided!limits!and!see!if!they!are!equal.!
x→1
x −1
!
1− x
!