University:
Washington State University
Course:
MATH 456 | Introduction to Statistical Theory
Academic year:

2023
Views:

82

Pages:

5
Author:

NeatGold

PageLIMITSIt is used to find the output ofa function as a value 15 approachedFor example let f(x) = x2 - a2 - 3If x = 3, the output is % = undefinedIf the value approaches 3 x3Thus, Lim f(x) = x2-9 =+3)x-3x-3Ex3)lim f(x) = x + 3 =6x-3If x approaches 3 from the left handlimit, , it is written as limf(x) = 6x-3-if it appraches 3 from the right handlimit it is written as lim f(x) = 6.2-3+x2.9 2.99 3 3.01 3.1f(x)5.9 5.99 6 6.01 6.1LHLRHLleft hand limitLights hand limitNote that the limit of a functionproduces an exact value and not anapproaching value.note RHL = f(a+) = n-a+ lim f(x) = ling f(a+h)LHL = f(a-) = x-a- lim f(x) = lim.f(a-h)and The limit exists only if LHL = RHLna lim f(x) = 2 him -a+ f (x) - x- lim a (x) = f(a+) OR Ha) DatePageGraphically x Lim 3 t (x) can be seen as follows -8RHD?65LUL3I-2,234>5Indeterminate FormThevaluesofindeterminateformscannotbe determined upfrout.The 7 forms are 0 , 00 + 00 x 0 8 - 00,100 00°, 0°000note%exact 0Theformsignifies0and not exactOandsoonfortheotherstooif thelimitisdeterminate(notanyofthe7formsabove),wecanusedirectSubstitutionBinomial ShortcutIf (7 + x) is part of the limit wherexD ,thenit can be written as (1+ nx,Usingthislim xn -ah = naxx-a When x-00AS x00 / 1/x 1/x21/x30when a < 0himaxasa=Thusx80afune ONE at1a = Imany pointsAtrickwhilesolvingthese questionsistoreplacex = 1/ywhere yO.Trigo LimitssinoSQtanoCO E [0, T/2 )1vi0upIsino COSDI> sino >> cosoQLim sindLim costI>0-20 Q0-00lim sino1>>10-20oThustimsind=I0-0oSimilartotheabove,thefollowinglimitscanproved-behim tanolimit sin (o-x)=I=O 0ooaoat1tim tanlo -x)lim sino0x=10-0 Q =Io -oc Expansouse'= 1 + x + xx/21 + x/31.00e-n = I - x + xx/24 - x3/31+ 00a' = I + xlna + (xima) 21 + (xina)++ 00sink = imaginary part of ein= x - x1/31 + x5/51 -+wwsx = real part of ein= 1- x1/21 + x4/41 - x6/618tank == x + x3/3 + 225/15++00in(1+x) = x-x-12 + x3/3 - x 1/4+(-laxsl)Ln(1-x) = -2-21/2-2 x3/3 - x4 14 + (-152<1)(1+2)" = e( 1- x/2 + 11x2/24 +)Log and Exp Limitshim 'n(1+x) = 1x->0xlim e*-1 = Ihim a*-1 =x-so xlnax-ong(x)tim g(2)(f(x)-1 1]aro um (1+x) th = ex-a Lim t(x) = eg(x)g(x) = lim e f(x) should be form100x-0q(x)t(x) shd be 0°or 00° form ePagenote If f (x) + g(x) exists, it does NOTimply that f(x) and /or g(x) existThe same goes for f(x) X g(x)L Hopital RuleIf lim t (x) is a000form.thenn a g(x)it is equivalenttoLim f((x).xag'(x)

Limits Lecture

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