Post on 25-Jun-2020
Topics in MathematicsPractical Session 1 - Limits
Walheer Barnabe
Walheer Barnabe Topics in Mathematics Practical Session 1 - Limits
Outline
(i) What is a limit?(ii) A rigorous definition of limits(iii) One-sided limits(iv) Known limits(v) Rules for limits(vi) Link with continuity(vii) Link with derivability(viii) Continuity & Derivability(ix) Limits at Infinity(x) Asymptotes(xi) Indeterminate cases(xii) L′Hospital′s rule(xiii) Sandwich Theorem(xiv) Series
Walheer Barnabe Topics in Mathematics Practical Session 1 - Limits
What is a limit?
f : R→ R : x → f (x)
limx→a
f (x) = A means that we can make f (x) as close to A as we want,
for all x sufficiently close to (but not equal to) a
The distance between two numbers can be measured by theabsolute value of the difference between them. Using absolutevalues, the definition can be reformulated in this way:
limx→a
f (x) = A means that we can make |f (x)−A| as small as we want,
for all x 6= a with |x − a| sufficiency small
Walheer Barnabe Topics in Mathematics Practical Session 1 - Limits
A rigorous definition of limits
We say that f (x) has limit (or tends to) A as x tends to a, and write
limx→a
f (x) = A , if for each number ε > 0 there exists a number δ > 0
such that |f (x)− A| < ε for every x with 0 < |x − a| < δ
(+graph)
What about f : Rn → R?
|x − a| becomes d(x , a) (Euclidean distance)
Walheer Barnabe Topics in Mathematics Practical Session 1 - Limits
One-sided limits
limx→a+ f (x) = A1 and limx→a− f (x) = A2
0 < |x − a| < δ or x ∈ (a− δ, a + δ) is replaced by (δ > 0):
(i) x ∈ (a− δ, a) for limit from the left
(ii) x ∈ (a, a + δ) for limit from the right
Remark: (a− δ, a) ∪ (a, a + δ) = ?
Walheer Barnabe Topics in Mathematics Practical Session 1 - Limits
Known limits
(i) limx→ak = k (a ∈ R)
(ii) limx→ax = a (a ∈ R)
(iii) limx→aex = ea (a ∈ R)
(iv) limx→a ln x = ln a (a ∈ R+0 )
(v) limx→a√x =√a (a ∈ R+)
(vi) limx→a sin x = sin a (a ∈ R)
(vii) limx→a cos x = cos a (a ∈ R)
(viii) limx→+∞ex = +∞ and limx→−∞ex = 0
(ix) limx→+∞ ln x = +∞ and limx→0+ ln x = −∞
(x) limx→+∞√x = +∞ and limx→0+
√x = 0
Walheer Barnabe Topics in Mathematics Practical Session 1 - Limits
Rules for limits
If limx→af (x) = A and limx→af (x) = B (with f and g defined inthe neighborhood of a (but not necessery at a)), then
(i) limx→a(kf (x)± hg(x)) = kA± hB(k , h ∈ R)
(ii) limx→a(f (x).g(x)) = A.B
(iii) limx→af (x)g(x) = A
B (if B 6= 0)
(iv) limx→a[f (x)]r = Ar (if Ar is defined and r is any real number)
Limits of composed of functions: limx→a g(f (x)) = limy→b g(y)How to find b? Compute limx→a f (x) = b
Walheer Barnabe Topics in Mathematics Practical Session 1 - Limits
Some examples
(1) limx→2(ex + 4x)
(2) limx→0+(4 + ln x)
(3) limx→0−ex
x
(4) limx→+∞ ex2
(5) limx→1
√3x + 1
(6) limx→2x3−5x
x2−5x+6
!!! You can not divide by 0, only by 0+ and 0−
Walheer Barnabe Topics in Mathematics Practical Session 1 - Limits
Link with continuity
f : R→ R : x → f (x), a ∈ domf
f is continous at x = a if limx→a
f (x) = f (a)
This is equivalent to the following three conditions:
(i) The function f must be defined at x = a(ii) The limit of f (x) as x tends to a must exist(iii) This limit must be exactly equal to f (a)
Same definition for f : Rn → R but sometimes limits more difficultto compute, a = (a1, . . . , an).
Walheer Barnabe Topics in Mathematics Practical Session 1 - Limits
One-sided continuity
f is left continous at x = a if limx→a−
f (x) = f (a)
f is right continous at x = a if limx→a+
f (x) = f (a)
This means that: f is continuous in a↔ f is right continuous in aand f is left continuous in a.
Walheer Barnabe Topics in Mathematics Practical Session 1 - Limits
Some useful results on continuous functions:
If f and g are continuous at a, then
(i) f + g and f − g are continous at a
(ii) fg and f /g (if g(a) 6= 0) are continous at a
(iii) [f (x)]r is continuous at a if [f (a)]r is defined
(iv) If f is continuous and has an inverse on the interval I , then itsinverse f −1 is continuous on f (I )
Walheer Barnabe Topics in Mathematics Practical Session 1 - Limits
Some examples
Study the continuity of the following functions:
(1) f (x) = x4+3x2−1(x−1)(x+2)
(2) f (x) =
{x2 − 1, for x ≤ 0−x2, for x > 0
(3) f (x) =
ex , for x ≤ 01 + x , for 0 < x ≤ 13− x , for x > 1
(4) f (x , y) = x2
x2+y2 (use polar coordinates, that is x = ρ cos θ and
y = ρ sin θ)
(5) f (x , y , z) = e√
x2+y2
z
Walheer Barnabe Topics in Mathematics Practical Session 1 - Limits
Link with derivability
f : R→ R : x → f (x), a ∈ domf
f is differentiable at x = a if limx→a
f (x)− f (a)
x − a∈ R [= f ′(a)]
f is left differentiable at x = a if limx→a−
f (x)− f (a)
x − a∈ R [= f
′l (a)]
f is right differentiable at x = a if limx→a+
f (x)− f (a)
x − a∈ R [= f
′r (a)]
This means that: f is derivable in a↔ f is right derivable in a andf is left derivable in a ↔ f
′l (a) = f
′r (a) = f ′(a).
Walheer Barnabe Topics in Mathematics Practical Session 1 - Limits
Link with derivability
f : Rn → R : x → f (x), a = (a1, . . . , an) ∈ domf
f is partial differentiable at xi = ai if
limxi→ai
f (a1, . . . , ai−1, xi , ai+1, . . . , an)− f (a)
xi − ai∈ R [=
∂f
∂xi(a)]
gradf (a) = ∇f(a) = ( ∂f∂x1
(a), . . . , ∂f∂xn
(a))
Walheer Barnabe Topics in Mathematics Practical Session 1 - Limits
Link with derivability:
f is differentiable at x = a if all the partial derivatives exist in a
↔
∇f(a) ∈ R
Walheer Barnabe Topics in Mathematics Practical Session 1 - Limits
Some examples
Study the derivability of the following functions (knowing that theyare continuous on their domains):
(1) f (x) =
{(x + 1)2, for x ≥ 02x + 1, for x < 0
(2) f (x) =
x2 + 1, for x < 07x + 1, for 0 ≤ x < 22x + 11, for x ≥ 2
(3) f (x , y) =
{ xyx2+y2 , for (x , y) 6= (0, 0)
0, for (x , y) = (0, 0)
(4) f (x , y , z) = 3+x2y−z3x2+1
in (0, 1, 2)
(5) f (x , y) = e−(x2+y2) in (0, 0) and (1, 2)
Walheer Barnabe Topics in Mathematics Practical Session 1 - Limits
Continuity & Derivability
If f is differentiable at x = a, then f is continuous at x = a
(+Proof for n = 1)
Derivable → ContinuousNot Continuous → Not DerivableContinuous → Derivable?
Walheer Barnabe Topics in Mathematics Practical Session 1 - Limits
Limits at Infinity
f (x)→∞ and g(x)→∞ as x → a, then
(i) f (x) + g(x)→∞ as x → a
(ii) f (x).g(x)→∞ as x → a
(iii) f (x)− g(x)→? as x → a
(iv) f (x)/g(x)→? as x → a
(+sign!)
Walheer Barnabe Topics in Mathematics Practical Session 1 - Limits
Asymptotes
f has a left vertical asympote of equation x = a ifflimx→a− f (x) = ±∞
f has a right vertical asympote of equation x = a ifflimx→a+ f (x) = ±∞
f has a vertical asympote of equation x = a iff limx→a f (x) = ±∞
f has a left horizontal asympote of equation x = b ifflimx→−∞ f (x) = b
f has a right horizontal asympote of equation x = b ifflimx→+∞ f (x) = b
f has a horizontal asympote of equation x = b ifflimx→±∞ f (x) = b
Walheer Barnabe Topics in Mathematics Practical Session 1 - Limits
Asymptotes
f has a right oblique asympote of equation y = ax + b ifflimx→+∞
f (x)x = a and limx→+∞ f (x)− ax = b
f has a left oblique asympote of equation y = ax + b ifflimx→−∞
f (x)x = a and limx→−∞ f (x)− ax = b
f has a oblique asympote of equation y = ax + b ifflimx→±∞
f (x)x = a and limx→±∞ f (x)− ax = b
Horizontal asymptote → no oblique asymptoteNo horizontal asymptote → oblique asymptote?
Walheer Barnabe Topics in Mathematics Practical Session 1 - Limits
Some examples
Study the asymptotes of the following functions:
(1) f (x) = e1/x
(2) f (x) = 1 + e−x2
(3) f (x) =
{1x , for x < 01 + e−x for x ≥ 0
(4) f (x) = x2−32x2+3x+1
(5) f (x) = 3x2−5x+62x−3
(6) f (x) = xe1/x
Walheer Barnabe Topics in Mathematics Practical Session 1 - Limits
Indeterminate cases
0
0,∞∞,∞.0,∞−∞, 1∞, 00,∞0
Tricks:
(1) limx→a f (x).g(x) = limx→af (x)
1g(x)
= limx→ag(x)
1f (x)
(2) limx→a[f (x)]g(x) = e ln limx→a[f (x)]g(x)= e limx→a g(x) ln f (x)
(3) limx→a(f (x)− g(x)) = limx→a
1g(x)− 1
f (x)1
f (x).g(x)
(4) etc.
Walheer Barnabe Topics in Mathematics Practical Session 1 - Limits
L′Hospital′s rule
Suppose that f and g are differentiable in the interval (α, β) thatcontains a, except possibly a, and suppose that f (x) and g(x)both tend to 0 as x tends to a. If g ′(x) 6= 0 for all x 6= a in (α, β),
and if limx→af ′(x)g ′(x) = L, then
limx→a
f (x)
g(x)= lim
x→a
f ′(x)
g ′(x)= L
This is true wheter L is finite, ∞ or −∞.
Walheer Barnabe Topics in Mathematics Practical Session 1 - Limits
Some examples
Compute the following limits:
(1) limx→+∞ln xx
(2) limx→+∞ x .e1x − x
(3) limx→0+(1 + 1x )x
(4) limx→0+ x ln x
(5) limx→0+ xe1/x
Walheer Barnabe Topics in Mathematics Practical Session 1 - Limits
Sandwich Theorem
Let f , g and h be three functions such that:
(1) limx→a f (x) = b and limx→a h(x) = b
(2) f (x) ≤ g(x) ≤ h(x) in the neighbordhood of a (possibly excepta)
then, limx→a g(x) = b
Very useful for e.g. trigonometric functions
Examples: limx→+∞ x + sin x , limx→+∞sin2 xx and
lim(x ,y)→(0,0)x2√x2+y2
Walheer Barnabe Topics in Mathematics Practical Session 1 - Limits
Series
Similar as before but n ∈ N, Un : N→ R : n→ Un
Notion of convergence: limn→+∞ Un ∈ R
More in the theoretical lectures.
Walheer Barnabe Topics in Mathematics Practical Session 1 - Limits