topics in mathematics practical session 1 - limits › 2015 › 10 › practical-ses… · walheer...

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


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


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)


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) = ?


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


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


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−


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).


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.


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 )


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


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).


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))


Link with derivability:

f is differentiable at x = a if all the partial derivatives exist in a

↔

∇f(a) ∈ R


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)


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?


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!)


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


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?


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


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.


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 −∞.


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


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


Series

Similar as before but n ∈ N, Un : N→ R : n→ Un

Notion of convergence: limn→+∞ Un ∈ R

More in the theoretical lectures.


topics in mathematics practical session 1 - limits › 2015 › 10 › practical-ses… · walheer...

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