Math Analysis I - Limits of Functions Part 5 (Landau Symbols)

Math Analysis Limits of Functions

Xo= ,Proof off)f . 0191 E1 . aXoi nt g -- ,18¥ 1 g)- o f fI 0cg)o h f .=I g x # Iii) 01181,(f) (g) Eo x ↳o =Proofoff)- o l d 01910111 EaXOi n= - . g -- ,I18¥0181 o u t=s g (jog)off.) 0cg) - O IX i2.

-0cg)d a t 01g)o(g)f - x # I,Proofd s lgt¥-¥'t 0¥84,;oft.FI#ools1tolgldi) - 01 5 10cg)041 t=g - x # Iid f-ocgl, g-ol ul, Ix . → - d u ) ][OCT)f - d u ) , X # IProof ¥-I1 - → oh h f - d u ) ,I - • EX v(i)u - fe e l ) yet(six) ICHe. a sg. →o - I I I ) )(ICE) (x-E),O X # Io=(TAI-ICE))Indeed, 0I-0111×1-1=2,.N¥¥¥ old,→T C H -T C H I→x .- o - o ( x - E ) ,olt¥¥t#= 0111×1-ICE) Ix .out →2.

CalcuevsoflimitsthroughLendewsymbotsFundementeelimits1 01 × 1,S i u X tx = x → osi¥-aIndeed, toll), t o× (x)oh)s i n xx x t ot XX ' →= o- ,l-¥2- Xo(x4 ' →C o s otx - ,Indeed,

'¥1 {toll) ↳X o= ,III - c o s 1×2) ' →X o=x t o ,l-¥ (XL)6 5 x - X ' →t oo ,3- t e n tocxlx x # ox - ,tq¥Indeed, t e l l l X t oI= ,t e u x (i) 01×1, XXX tt = ' →X o o=ext (x)I l X i a ot t oX ,e¥Indeed, toll) X t o- 1 ,E - I ex(x) (x),I Xt t oX t o = X i a o=2. Math Analysis Limits of Functions 12 of 315- ( i t XIlog 01 × 1 X i a otx= ,lojf#=Indeed (i), t ot o ×1log ( I t x) 01 × 1 t otx X= ,&( I t x)6- t t a x )l x x X i a o= ,x)'(It - ,Indeed (i) Xt A oh o× - t ,& -x)(l 4 (x),It t o Xx t o=x)"(1 o(x),t ' →t X= ti x x oI (x)hs i 01 × 1 ' →xn xt o= ,sighIndeed out, ' →xi t o-hs i (x)u Xx t o t o= ,¥1 (x) I t ocx4h →xc o s t o- ,6511¥Indeed, I t out, X i a o=I z 01×2),(x) I6 5 4 X →t t o= 01×1,I (x)t e n h xtx ↳ o=Indeed tex t toll), xl ' → o=t e n ok),4 X i a oX t=2. Math Analysis Limits of Functions

13 of 31

Theorem (pnincipeeoswegle-g.by)

IR, IRE IE,E E EF, G,c f , g: →(G),off),f IXg - O ↳=then, FLim Eft e i n=tagE+ a→end d o e s n't l i m i t st w ot h ei f one ofdoesnexist, t h e' t o t h e rs oProof 2nd1stI.lt#xoFat = 8/910I t2ndthen t h e d o e s n o tfactorl i m i ttheaffect2. Math Analysis Limits of Functions 14 of 31wttsiuhn5t#=l i me . g.. →u t o o 425n e log u - 2ttu 7 t o C u 7 l t # ILinz= =2T)421T ( u (nut) (n't)t to t• on" (ut)e i n t o- - (nut)42'Tt o o↳n t oby t h e principle negeegibility,o fthe l i m i t equivalent t ogiven i sn't l i m o ot=¥4 h i , s i n x t x t t . t tlion =• × → 0 x - l - x 2t e n c o s htx ¥t I t01×1 t o l d ) olx)t t xx- I t ON¥N 01 × 21t oX t i t ¥=o(x),01 × 21=01 × 1 e n dsince (x) t a x )(x)t oX 01 × 1t ot X t- (x)ok) t o l x )(x)X t o tt o01 × 12 tX= t oX,- O Nt× (x) ¥¥=z x t oby 2,n e g l i g i b i l i

t y, x - t oI t o =,2. Math Analysis Limits of Functions 15 of 31

Asymptotes

Consider a function f defined in a neighbourhood of +∞ and study its behaviour for x → +∞. A remarkable case is that in which f behaves as a polynomial of first degree. Geometrically speaking, this corresponds to the fact that the graph of f will more and more look like a straight line.

Precisely, we suppose there exist two real numbers and such that

ASYMPTOTE

t h a t fliessay e nw ef o r ifX i e > t o o (unity))( I N -lim - Ox . o o→ fix) di), ' →M x t X t o ot y=

We then say that the line g(x) = mx + q is a right asymptote of the function f.

X u - o ofor

LEFT1 H O R I Z O N TA L

saidif t i sY- M Xm-o, it A S Y M P TOT E2- seidi f OBLIQUEi si n f o , y - n e x t y A S Y M P TOT E

A1 ¥2. Math Analysis Limits of Functions 16 of 31

The asymptote’s coefficients can be determined by the following relations:

Moreover, if the function consists of the ratio of two polynomials, with

The numerator showing a higher degree with respect to the denominator, then the asymptotes can be determined by means of polynomials division.

2. Math Analysis Limits of Functions 17 of 31

Remark

The definition of (linear) asymptote given is a particular instance of the following:

If f(x) → a as x → ∞, then we say that y = mx + q is an oblique asymptote.

Definition (vertical asymptote): If the line is an oblique or horizontal asymptote for both x → ∞ and x → -∞, we say that it is a complete oblique or complete horizontal asymptote for f.

Definition (vertical asymptote): If the distance between points on the graph of f and on a vertical asymptote with the same y-coordinate converges to zero as x → a, then we say that x = a is a vertical asymptote for f.

If the limit condition holds only for x → a+ or x → a-, we talk about a

vertical right or left asymptote respectively.

2. Math Analysis Limits of Functions 18 of 31

Find the asymptotes of the function:

-X^3/(t*l*x^3/t^2 + (x*f(x))^5)

f(x) = (x^2*3 + 2*2*t)/(x - l) = -2*5*t*x/(x^2*t + 5*x - l)

Asymptote is x = -2/5

3. Math Analysis Limits of Functions 19 of 31

Find the asymptotes of the function:

-log(x^2 - l)/(x - l) = -log(x^2 - l)/(x - l)

Asymptote is y = 2

4. Math Analysis Limits of Functions 20 of 31

Find the asymptotes of the function:

log(x^3 - t)/(x - l) = log(x^3 - t)/(x - l)

Asymptote is x = 0

→Fundamentalists1 01 × 1,S i u X tx = X i a ox - X ,s i n ' →X oI-I2- XoCx4 ' →C o s otx = ,l-¥ l - l )c o s ' →Xx on ,3- t e n tocx)x x # ox - /t e n x →xx , onex- ok)1 1 X i a ot x t ,l-l)e × N i t X i a ox ,2. Math Analysis Limits of Functions 21 of 315- ( i t XIlog (x) X i a otx= o ,x)log 1 I t ~ x t ox,&( I t x)6- t (x)l t ox x X i a o= ,XP(I t N l x ' →t x x , oI (x)hs i 01 × 1 ' →xn xt o= ,s i n u x x →x~ o,¥1 (x) I t ocx4h →xc o s t o= ,It} 1=11c o s h x ' →~ x o,01×1,I (x)t e n h xtx ↳ o=t e n w x w xx , ' → o× 'Tsink Xt t c o s tx - iL i me . g . - t e n x - l - X -t u s hX → o × (x),× 'Tsince sinx ZX tXt t c o s tx - i = o× 'Tsink X tt c o s tx - I 2~ xx - l -X- ~a n a l o g o u s l y, t e n t u s h x× × 'T ¥sink Xt t c o s +x - iLim fiefthen, =t ¥ ¥ ×* → o2. Math

Analysis Limits of Functions 22 of 31 (principle of substitution)

Theorem: If f is a function and g is a function such that g approaches L as x approaches ∞, then f(g(x)) approaches F as x approaches L.

Proof: Let I be an interval containing L, and let x approach L. Then, by definition, there exists a number δ such that for all x in the interval I except possibly at L, we have |x - L| < δ.

Since g(x) approaches L as x approaches ∞, there exists a number N such that for all x greater than N, we have |g(x) - L| < δ.

Now, consider the function f(g(x)). Let ε be a positive number. We want to show that there exists a number M such that for all x greater than M, we have |f(g(x)) - F| < ε.

Since f is continuous at L, there exists a number ε' such that for all y in the interval I except possibly at L, we have |f(y) - F| < ε'.

Let M be a number greater than N such that for all x greater than M, we have |g(x) - L| < δ. Then, for all x greater than M, we have |f(g(x)) - F| < ε'.

Therefore, we have shown that for any positive number ε, there exists a number M such that for all x greater than M, we have |f(g(x)) - F| < ε. This proves that f(g(x)) approaches F as x approaches L.