Chapter 4 : Limit of a function€¦L L G Advanced Math and Science Pilot Class Paris – Abu Dhabi Mathematics, Grade 12 2016 – 2017 Chapter 4 : Limit of a function

L L G Advanced Math and Science Pilot Class

Paris – Abu Dhabi Mathematics, Grade 12

2016 – 2017

Chapter 4 : Limit of a function

The aim of this chapter is to study the behaviour of a function f at the endpoints of the intervals of its domain.

Karl Theodor Wilhelm Weierstrass (1815 – 1897) was a German mathematician who is often cited as the "father of modern analysis". In the whole chapter, we will call 𝑎, either −∞ or +∞ or a real number. f is a function defined in a neighborhood of 𝑎.

I- Definitions :

We say that the limit in 𝑎 of 𝑓 is +∞ when for any real number A (as great as we want), 𝑓(𝑥) is greater than 𝐴

as 𝑥 approaches 𝑎. We denote lim𝑥⟼𝑎

𝑓(𝑥) = +∞.

We say that the limit in 𝑎 of 𝑓 is −∞ when for any negative number A (as low as we want), 𝑓(𝑥) is smaller

than 𝐴 as 𝑥 approaches 𝑎. We denote lim𝑥⟼𝑎

𝑓(𝑥) = −∞.

We say that the limit in 𝑎 of 𝑓 is l when for any real number 𝜀 (as small as we want), 𝑓(𝑥) belongs to the

interval ] l—𝜀 ; l +𝜀[ as 𝑥 approaches 𝑎. We denote lim𝑥⟼𝑎

𝑓(𝑥) = l.

Remark : lim𝑥⟼𝑎

𝑓(𝑥) = l lim𝑥⟼𝑎

𝑓(𝑥) − l= 0.

Attention : Not all the functions have limits : see 𝑓(𝑥) = cos 𝑥 in +∞.

Limits you should know :

Square function

lim𝑥→+∞

𝑥2 = +∞

lim𝑥→−∞

𝑥2 = +∞

Reciprocal function

lim𝑥→+∞

1

𝑥= lim

𝑥→−∞

1

𝑥= 0

lim𝑥→0+

1

𝑥= +∞

lim𝑥→0−

1

𝑥= −∞

Cubic Function

lim𝑥→+∞

𝑥3 = +∞

lim𝑥→−∞

𝑥3 = −∞

Square root function

lim𝑥→+∞

√𝑥 = +∞

!!! Saying that 𝐥𝐢𝐦𝒙⟼+∞

𝒇(𝒙) = +∞ does not necessarily means that 𝒇 is increasing.

Graphic interpretation :

If 𝑎 is a real number and lim𝑥→𝑎

𝑓(𝑥) = ±∞, then the straight line which equation is 𝑥 = 𝑎 is a vertical

asymptote for 𝐶𝑓.

If 𝑏 is a real number and lim𝑥→±∞

𝑓(𝑥) = 𝑏, then the straight line which equation is 𝑦 = 𝑏 is a horizontal

asymptote for 𝐶𝑓.

If 0)(lim

baxxfx

then the line with equation 𝒚 = 𝒂𝒙 + 𝒃 is an oblique asymptote of the curve Cf in

. The same in - .

From a graphic point of view, the curve of 𝒇 indefinitely approaches its asymptote but never reaches it.

Attention : Not all the curves representing functions have asymptotes (see parabolas).

II- Calculating limits :

𝑎 is either −∞ or +∞ or a real number, l and l’ are real numbers.

Sum

𝐼𝑓 lim𝑥→𝑎

𝑓(𝑥) = l l l +∞ −∞ +∞

𝐴𝑛𝑑 lim𝑥→𝑎

𝑔(𝑥) = l' +∞ −∞ +∞ −∞ −∞

𝑇ℎ𝑒𝑛 lim𝑥→𝑎

[𝑓(𝑥) + 𝑔(𝑥)] = l + l’ +∞ −∞ +∞ −∞ IF

Product

𝐼𝑓 lim𝑥→𝑎

𝑓(𝑥) = l l < 0 l> 0 l +∞ −∞ +∞

𝐴𝑛𝑑 lim𝑥→𝑎

𝑔(𝑥) = l' +∞ +∞ 0 0 −∞ −∞

𝑇ℎ𝑒𝑛 lim𝑥→𝑎

[𝑓(𝑥) × 𝑔(𝑥)] = l × l’ −∞ +∞ 0 IF +∞ −∞

Quotient

lim𝑥→𝑎

𝑓(𝑥) = l l l > 0 0 0 0 +∞ ∞

lim𝑥→𝑎

𝑔(𝑥) = l'≠ 0 +∞ 0+ 0 l ≠ 0 +∞ 0+ ∞

lim𝑥→𝑎

[𝑓(𝑥)

𝑔(𝑥)] =

𝑙

𝑙′ 0 +∞ IF 0 0 +∞ IF

Examples : Evaluate the following limits :

lim𝑥→+∞

2

𝑥 − 1 ; lim

𝑥→1

2

(𝑥 − 1)2 ; lim

𝑥→1

2

𝑥 − 1 ; lim

𝑥→3

𝑥 − 3

2𝑥 + 5 ; lim

𝑥→4

−𝑥 + 2

𝑥2 + 5 ; lim

𝑥→−1

5𝑥 − 3

3𝑥2 + 2𝑥 − 1

Particular cases :

o The limit in +∞ and −∞ of a polynomial function is the limit of its leading term .

o The limit in +∞ and −∞ of a rational function is the limit of the ratio of its leading terms.

Examples : Evaluate the following limits :

lim𝑥→+∞

−4𝑥5 + 3𝑥4 +1

2𝑥3 + 10 ; lim

𝑥→−∞−𝑥7 − 123𝑥5 −

5

6𝑥4 + 10

lim𝑥→+∞

3𝑥4 + 5𝑥2 − 𝑥 − 15

2𝑥4 + 5𝑥3 − 3𝑥2 + 𝑥 − 35 ; lim

𝑥→−∞

3𝑥3 + 5𝑥2 − 𝑥 − 15

2𝑥4 + 5𝑥3 − 3𝑥2 + 𝑥 − 35 ; lim

𝑥→+∞

3𝑥6 + 5𝑥2 − 𝑥 − 15

2𝑥4 + 5𝑥3 − 3𝑥2 + 𝑥 − 35

III- Limits and inequalities :

𝑙 and 𝑙′ are two real numbers. If for any 𝑥 in a neighborhood of 𝛼 we have :

1/ )()( xgxf and

)(lim xgx

then lim𝑥→𝛼

𝑓(𝑥) = +∞

2/ )()( xgxf and

)(lim xgx

then lim𝑥→𝛼

𝑓(𝑥) = −∞

3/ )()( xulxf and 0)(lim

xux

then lim𝑥→𝛼

𝑓(𝑥) = 𝑙

4/ )()()( xvxfxu and lxvxuxx

)(lim)(lim

then lim𝑥→𝛼

𝑓(𝑥) = 𝑙

5/ )()( xgxf , lxfx

)(lim

and ')(lim lxgx

then 𝑙 ≤ 𝑙′

Examples : Using the previous properties evaluate the given limit. Check with your calculator.

𝑎 = lim𝑥→+∞

𝑥 + sin 𝑥 𝑏 = lim𝑥→−∞

𝑥 + sin 𝑥 𝑐 = lim𝑥→+∞

(sin 𝑥

𝑥) 𝑑 = lim

𝑥→0 𝑥 sin (

1

𝑥) 𝑒 = lim

𝑥→+∞(

𝑥+sin 𝑥

2𝑥+1)

IV- Composition of two functions :

Let 𝑓 and 𝑔 be two functions : 𝑓 defined on 𝒟𝑓 and 𝑔 on 𝒟𝑔.

The function ℎ = 𝑓 ∘ 𝑔 is defined on 𝒟 = {𝑥 ∈ 𝒟𝑔, 𝑔(𝑥) ∈ 𝒟𝑓} by ℎ(𝑥) = 𝑓 ∘ 𝑔(𝑥) = 𝑓(𝑔(𝑥)).

𝒟 ⊂ 𝒟𝑔 ⟶ 𝒟𝑓 ⟶ ℝ

𝑔 ∶ 𝑥 ⟼ 𝑔(𝑥)

𝑓 ∶ 𝑋 ⟼ 𝑓(𝑋)

ℎ = 𝑓 ∘ 𝑔 ∶ 𝑥 𝑔(𝑓(𝑥))

Theorem : Let 𝑓, 𝑔, ℎ be 3 functions such as ℎ = 𝑓 ∘ 𝑔.

𝑎, 𝑏, ℓ are real numbers or infinite. If lim𝑥→𝑎

𝑔(𝑥) = 𝑏 and lim𝑥→𝑏

𝑓(𝑥) = ℓ then lim𝑥→𝑎

𝑓 ∘ 𝑔(𝑥) = ℓ.

Examples : Evaluate the following limits : lim𝑥→+∞

√4𝑥+2

𝑥−1 ; lim

𝑥→+∞sin (

1

𝑥)