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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)