Limit Formulas
10.1 Definition of Limit
LIMIT OF A FUNCTION (INFORMAL DEFINITION)
The notation
x!climfxDL
is read ‘‘the limit off(x) asxapproachescisL’’ and means that the functional valuesf(x) can be made arbitrarily close toLby choosingxsufficiently close toc.
LIMIT OF A FUNCTION (FORMAL DEFINITION)
The limit statement
x!climfxDL
means that for each >0, there corresponds a number υ >0 with the property that
jfxLj< whenever 0<jxcj< υ
A FUNCTION DIVERGES TO INFINITY (INFORMAL DEFINITION)
A function fthat increases or decreases without bound as xapproachescis said todiverge to infinity1atc. We indicate this behavior by writing
(continued)
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x!climfxD C1 ifxincreases without bound and by
x!climfxD 1 ifx decreases without bound.
INFINITE LIMIT (FORMAL DEFINITION)
We write limx!cfxD C1 if, for any number N >0 (no matter how large), it is possible to find a numberυ >0 such thatfx > Nwhenever 0<jxcj< υ.
LIMITS INVOLVING INFINITY
The limit statement limx!C1fxDLmeans that for any number >0, there exists a numberN1such that
jfxLj< wheneverx > N1
for x in the domain of f. Similarly limx!1fxDM means that for any >0, there exists a number N2 such that
jfxMj< wheneverx < N2
LIMIT OF A FUNCTION OF TWO VARIABLES (INFORMAL DEFINITION)
The notation
x,y!xlim0,y0fx, yDL
continued
means that the functional values f(x, y) can be made arbitrarily close to Lby choosing the point (x, y) close to the pointx0, y0.
LIMIT OF A FUNCTION OF TWO VARIABLES (FORMAL DEFINITION)
Suppose the point P0x0, y0 has the property that every disk centered atP0contains at least one point in the domain offother thanP0itself. Then the numberLis thelimit off atPif, for every >0, there exists aυ >0 such that jfx, yLj< whenever 0<
xx02Cyy02< υ In this case, we write
x,y!xlim0,y0fx, yDL
10.2 Rules of Limits
BASIC RULESFor any real numbers a and c, suppose the functions f and g both have limits at x Dc. Suppose also that both limx!C1fx and limx!1fxexist.
Limit of a constant
x!climk Dkfor any constantk
Limit ofx lim
x!cx Dc Scalar rule lim
x!c[afx]Dalim
x!cfx
Sum rule lim
x!c[fxCgx]D limx!cfxClim
x!cgx Difference rule lim
x!c[fxgx]D limx!cfxlim
x!cgx Linearity rule limx!C1[afxCbgx]D
alimx!C1fxCblimx!C1gx
Product rules lim
x!c[fxgx]D[lim
x!cfx][lim
x!cgx]
limx!C1[fxgx]D
[limx!C1fx] [limx!C1gx]
Quotient rules lim
x!c
fx gx D
x!climfx
x!climgx if lim
x!cgx6D0 limx!C1fx
gx D limx!C1fx
limx!C1gx if limx!C1gx6D0 Power rules lim
x!c[fx]n D
x!climfx n
nis a rational number
limx!C1[fx]n D[limx!C1fx]n Limit limitation
theorem
Suppose lim
x!cfxexists andfx½0 throughout an open interval
containing the numberc, except possibly atcitself. Then limx!cfx½0.
The squeeze rule Ifgx fx hxfor allxin an open interval containingc(except possibly atcitself) and if
x!climgxDlim
x!chx DL then lim
x!cfxDL.
Limits to infinity limx!C1 A
xn D0 and limx!1 A xn D0 Infinite-limit
theorem
If lim
x!cfxD C1and lim
x!cgx D A, then
x!clim[fxgx]D C1and lim
x!c
fx gx D C1ifA >0
x!clim[fxgx]D 1and lim
x!c
fx gx D 1ifA <0
l’Hopital’s ruleO Letfandgbe differentiable functions on an open interval containingc (except possibly atcitself).
If lim
x!c
fx
gx produces an indeterminate form0
0 or 1 1, then
x!clim gx Dlim
x!c g0x
provided that the limit on the right side exists.
TRIGONOMETRIC LIMITS
x!climcosxDcosc lim
x!csecx Dsecc
x!climsinxDsinc lim
x!ccscx Dcscc
x!climtanxDtanc lim
x!ccotx Dcotc
x!0lim sinx
x D1 lim
x!0
sinax
x Dalim
x!0
tanx
x D1 lim
x!0
1cosx x D0 MISCELLANEOUS LIMITS
n!C1lim
1C 1 n
n
De lim
n!01Cn1/nDe
n!C1lim
1C k n
n
Dek lim
n!C1p
1C 1 n
nt
Dpet
n!C1lim n1/nD1
10.3 Limits of a Function of Two Variables
BASIC FORMULAS AND RULES FOR LIMITS OF A FUNCTION OF TWO VARIABLESSuppose lim
x,y!x0,y0fx, y and lim
x,y!x0,y0gx, y both exist, with lim
x,y!x0,y0fx, yDLand lim
x,y!x0,y0gx, yDM.
Then the following rules obtain:
Scalar rule lim
x,y!x0,y0[afx, y]
Da lim
x,y!x0,y0fx, yDaL
Sum rule lim
x,y!x0,y0[fCg]x, y D
x,y!xlim0,y0fx, y
C
x,y!xlim0,y0gx, y DLCM
Product rule lim
x,y!x0,y0[fg]x, y D
x,y!xlim0,y0fx, y lim
x,y!x0,y0gx, y
DLM Quotient rule lim
x,y!x0,y0
f g
x, yD
x,y!xlim0,y0fx, y lim
x,y!x0,y0gx, y D L M ifM6D0
Substitution rule
Iff(x, y) is a polynomial or a rational function, limits may be found by substituting for x and y (excluding values that cause division by zero).
