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2.2. Limits of Functions - University of Manitoba

Limits of FunctionsWe now start the real content of the limit is a tool for describing how (real-valued) Functions behaveclose to a writelimx af(x) =Land say the limit off(x), asxapproachesa, equalsL ;if the values off(x) can be made as close as we like toLby takingxto be sufficiently close toa(on either side ofa) but not limit involves what is going on around a point, and does notcare what happens at will talk more soon about how to calculate the Limits is never the way to go in practice, but if you work throughthe guessing examples in the book, you may have a better idea ofwhat is going on and how Limits LimitsWritelimx a f(x) =Land say that theleft-hand limit off(x)asxapproachesais equal toLif we can make the values off(x) arbitrarily closetoLby takingxto be sufficiently close toaandxless a+f(x) =Land say that theright-hand limit off(x)asxapproachesais equal toLif we can make the values off(x) arbitrarily closetoLby takingxto be sufficiently close toaandxgreater ,limx 0 H(x) = 0andlimx 0+H(x) = 1.

2.2. Limits of Functions We now start the real content of the course. A limit is a tool for describing how (real-valued) functions behave close to a point. We write lim x→a f(x) = L and say “the limit of f(x), as x approaches a, equals L”; if the values of …

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Transcription of 2.2. Limits of Functions - University of Manitoba

1 Limits of FunctionsWe now start the real content of the limit is a tool for describing how (real-valued) Functions behaveclose to a writelimx af(x) =Land say the limit off(x), asxapproachesa, equalsL ;if the values off(x) can be made as close as we like toLby takingxto be sufficiently close toa(on either side ofa) but not limit involves what is going on around a point, and does notcare what happens at will talk more soon about how to calculate the Limits is never the way to go in practice, but if you work throughthe guessing examples in the book, you may have a better idea ofwhat is going on and how Limits LimitsWritelimx a f(x) =Land say that theleft-hand limit off(x)asxapproachesais equal toLif we can make the values off(x) arbitrarily closetoLby takingxto be sufficiently close toaandxless a+f(x) =Land say that theright-hand limit off(x)asxapproachesais equal toLif we can make the values off(x) arbitrarily closetoLby takingxto be sufficiently close toaandxgreater ,limx 0 H(x) = 0andlimx 0+H(x) = 1.

2 For any functionfand anya, the general limit limx af(x) existsand equalsLif and only if both the left-hand the right-hand limitsexist, andlimx a f(x) = limx a+f(x) =L:Infinite LimitsIn general, and formally, letfbe a function defined on both sidesofa, except possibly ataitself. Thenlimx af(x) = means that the values off(x) can be made arbitrarily large bytakingxsufficiently close toabut not equal ,limx af(x) = means that the values off(x) can be made arbitrarily large neg-ative by takingxsufficiently close toabut not equal and natural definitions can be made for one sided limitsas well:limx a f(x) = limx a+f(x) = limx a f(x) = limx a+f(x) = The Limit LawsSupposecis a constant and the Limits limx af(x) and limx ag(x)exist. Then, limx a[f(x) +g(x)] = limx af(x) + limx ag(x) limx a[f(x) g(x)] = limx af(x) limx ag(x) limx a[cf(x)] =climx af(x) limx a[f(x)g(x)] = limx af(x) limx ag(x) limx af(x)g(x)=limx af(x)limx ag(x)(if limx ag(x) = 0).

3 Limx a[f(x)]n= [limx af(x)] a polynomial or a rational function andais inthe domain off, thenlimx af(x) =f(a):This now allows us to actually find the limit of many (x) g(x) whenxis neara(except possibly ata) and the Limits offandgboth exist asxapproachesa, thenlimx af(x) limx ag(x):The Squeeze (x) g(x) h(x) whenxisneara(except possibly ata), andlimx af(x) = limx ah(x) =L;thenlimx ag(x) = ContinuityThe idea of continuity is basic: a function is continuous if it s graphcan be drawn without lifting your pencil off the paper. Howeverwe now formalize this:A functionfis said to becontinuous at a numberaiflimx af(x) =f(a):Notice that three things are implied by this definition for a functionfto be continuous (a) is defined, that is,ais in the domain off2. limx af(x) exists3. limx af(x) =f(a):If a function is not continuous ata, we say it isdiscontinuousata, orfhas a function is discontinuous, there are a number of special waysit can be functionfiscontinuous from the right at a numberaiflimx a+f(x) =f(a);andfiscontinuous from the left ataiflimx a f(x) =f(a):A function iscontinuous on an intervalif it is continuousat every number in the interval (iffis defined only on one sideof an endpoint of the interval, we understand continuous at theendpoint to mean continuous from the right or continuous fromthe left, as appropriate).

4 Continuous ataandcis a constant,then the following Functions are also continuous ata: f+g f g cf fg fg(ifg(a) = 0). polynomial is continuous everywhere, that is,on the whole real rational function is continuous at every pointin it s following types of Functions are continuous atevery number in their domains: Polynomials Rational Functions Root Functions Trigonometric Functions Inverse Trigonometric Functions Exponential Functions Logarithmic continuous ataandfis continuous atg(a),then the compositionf ggiven by (f g)(x) =f(g(x)) is con-tinuous (Intermediate Value Theorem)Suppose thatfis continuous on the closed interval [a;b] and letNbe anynumber betweenf(a) andf(b) wheref(a) =f(b). Then thereexists a numbercin (a;b) such thatf(c) = Limits at Infinity: Horizontal AsymptotesLetfbe a function defined on some interval (a; ).

5 Thenlimx f(x) =Lmeans that the values off(x) can be made arbitrarily close toLby takingxto be sufficiently ,limx f(x) =Lmeans that the values off(x) can be made arbitrarily close toLby takingxsufficiently large liney=Lis called ahorizontal asymptoteof the curvey=f(x) if eitherlimx f(x) =Lor limx f(x) = >0 is a rational number, thenlimx 1xr= 0:Ifr>0 is any rational number such thatxris defined for allx,thenlimx 1xr= 0 anyr Z+,limx 1xr= limx 1xr= 0:Infinite Limits at infinityThe notationlimx f(x) = means that the values off(x) become large asxbecomes , we define the other notations:limx f(x) = limx f(x) = limx f(x) = Derivatives and Rates of ChangeThe major purpose of Limits in this course is to be able to talkaboutinstantaneousrates of any two points, one can easily determine the average rateof change between them:Fix a functionf(x), and a pointP= (a;f(a)) on the we can find the rate of change fromPto any other pointQ= (x;f(x)) by the formula:mPQ=f(x) f(a)x a:Now we want to letQapproachP.

6 The rate of change will ap-proach closer and closer to the rate of the change of thetangentatP. Thetangent lineto the curvey=f(x) at the pointP(a;f(a)) is the line throughPwith slopem= limx af(x) f(a)x a:There is another version that is sometimes easier to use. Leth=x a:Thenx=a+h;so the same formula above becomesmPQ=f(x) f(a)x a=f(a+h) f(a)a+h a=f(a+h) f(a)hNow, we can rewrite the slope of the tangent line asm= limx af(x) f(a)x a= limh 0f(a+h) f(a) of a functionfat a numbera, denotedf (a), isf (a) = limh 0f(a+h) f(a)hif this limit derivative is also known as the instantaneous rate of changewith respect The derivative as a the last section, we considered the derivative of a functionfata fixed numbera:f (a) = limh 0f(a+h) f(a)h:Here we change our point of view and let the numberavary. Ifwe replaceain the above equation by a variablex, we getf (x) = limh 0f(x+h) f(x)h:For eachxfor which this limit exists, define the function thatmapsxto this numberf (x).

7 Thenf is a new function, calledthederivative functionfis calleddifferentiable ataiff (a) exists. It isdifferentiable on the open interval (a;b) [or (a; ), or ( ;a),or ( ; )] if it is differentiable at every number in the differentiable ata, thenfis continuous show thatfis continuous ata, it must be shown thatlimx af(x) =f(a):Sincefis differentiable ata,f (a) = limh 0f(a+h) f(a)h= limx af(x) f(a)x aexists. So,limx af(x) = limx a(f(a) +f(x) f(a))(add/subf(a));= limx af(a) + limx a(f(x) f(a))=f(a) + limx af(x) f(a)x a(x a)(mult/div byx a);=f(a) + limx af(x) f(a)x alimx a(x a)=f(a) +f (a) 0;=f(a):Thereforefis continuous the converse of this theorem is not true. We showed a fewslides ago thatf(x) =|x|is not differentiable at 0, butlimx 0|x|= 0 =f(0)and therefore|x|is continuous at sure you know the direction this theorem : How can a function look if it is not differentiable?


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