This was created by Keenan Xavier Lee - 2014. See my website for more information, lee-apcalculus.weebly.com.

14 Squeeze Theorem

Standards :

MCAZ

MCAZA

This was created by Keenan Xavier Lee - 2014. See my website for more information, lee-apcalculus.weebly.com.

o@ Calculating Limits Algebraically

aximuxI5edmnEEhxeIaxhnaxtt.1t1-2.WnstoEIx-FIieoknErxII.oEEI.tIxkIxExEk-Htxt2tt3.4niXt2-9-x-tEcxrIItxiFtETIEztI-lxmtxfI-rtI-EE.n@SqueezetheoremLetsconsiderxlingox2sinext.5ometimeswecannAfindlimittsdlieoHylbyalgebraicmanipulatioh.sothesqueezetheoremCorYandwiohtheoremYcanhe1p.5upposewehave2functionslrepresentedgraphically.9HhHgkWKtAls9letssaythereisafunoHnfcxjthatThMlNMisBandwiuhedbelweenthe2fundimsA.k

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This was created by Keenan Xavier Lee - 2014. See my website for more information, lee-apcalculus.weebly.com.

Fequeethremlfgcxtfcxtkforallxfainsomeintervalabouta,and

t±a94=tmaK'' =L ,

then limfkkl .

x a

Gangbacktothe previous situation .

lxinoxsinfg )

noteiweknowthatvaluesofsinefnnaionlie between - 1and1.

^

tntoxlgmjosinext Uusetthis ! Am. 1±x2sin#< 1 #-1×21×2sinkx )< 1×2

l '

×hm jx2=0 and,km ox2=o

Sqthesqulezetheoremgvexlimgox 's .in#=O .

This was created by Keenan Xavier Lee - 2014. See my website for more information, lee-apcalculus.weebly.com.

ExampleD If 3×441×3+2 for 01×12,evaluate xlignfcx) .

him 3x =3 (1) =3

÷xM 1×3 +2=43+2--3

Since both limits equal 3, then applying the Squeeze Theorem gives thatkin fcx) =3 .

x.ExampleD Find

lxingyfcx) given that 4 Ifk ) e X +6×-3 for all x .

him 4 = 4-

÷thtx 46×-3=(112+64) -3=4So

, ,Y;n±fcxI=4 when the squeeze theorem is applied .