Limits and Continuity

February 26, 2010

Limits

f : D → R, with D a subset of the plane.P0: point in plane such that:

◮ f is defined arbitrarily close to P0;

◮ f is not necessarily defined at P0.

Example:

f : D = R2 \ {P0(0, 0)} → R, f (x , y) =

x2y

x2 + y2

◮ Defined arbitrarily close to (0, 0);

◮ Not defined at P0(0, 0).

Limits

f : D → R, with D a subset of the plane.P0: point in plane such that:

◮ f is defined arbitrarily close to P0;

◮ f is not necessarily defined at P0.

Example:

f : D = R2 \ {P0(0, 0)} → R, f (x , y) =

x2y

x2 + y2

◮ Defined arbitrarily close to (0, 0);

◮ Not defined at P0(0, 0).

Question: What happens to f (Q) as Q gets closer to P0?

Example

f (x , y) =x2y

x2 + y2

f (Q) →? as Q → P0(0, 0)

Example

f (x , y) =x2y

x2 + y2

f (Q) →? as Q → P0(0, 0)

Numerical approach:

Q1(0.1, 0.1) =⇒f (Q1) = f (0.1, 0.1) ≃ 0.05

Q2(0.01,−0.02) =⇒f (Q2) = f (0.01,−0.02) ≃ −0.004

Q3(−0.003, 0.001) =⇒f (Q3) = f (−0.003, 0.001) ≃ 0.0009

Numerical data suggests:

Example

f (x , y) =x2y

x2 + y2

f (Q) →? as Q → P0(0, 0)

Numerical approach:

Q1(0.1, 0.1) =⇒f (Q1) = f (0.1, 0.1) ≃ 0.05

Q2(0.01,−0.02) =⇒f (Q2) = f (0.01,−0.02) ≃ −0.004

Q3(−0.003, 0.001) =⇒f (Q3) = f (−0.003, 0.001) ≃ 0.0009

Numerical data suggests:

f (Q) is closer and closer to 0 as Q → P0(0, 0)

Limit of a function at a point

f : D → R, with D a subset of the plane.P0: point in plane such that:

◮ f is defined arbitrarily close to P0;

◮ f is not necessarily defined at P0.

DefinitionA value L (finite or infinite) is the limit of f at P0 ifwe can keep the values of f (Q) as close to L as we wantby keeping Q close enough to P0, but not equal to P0.

Notation:

L = limQ→P0

f (Q) or L = lim(x,y)→(x0 ,y0)

f (x , y)

Limit of a function at a point

f : D → R, with D a subset of the plane.P0: point in plane such that:

◮ f is defined arbitrarily close to P0;

◮ f is not necessarily defined at P0.

DefinitionA value L (finite or infinite) is the limit of f at P0 ifwe can keep the values of f (Q) as close to L as we wantby keeping Q close enough to P0, but not equal to P0.

Notation:

L = limQ→P0

f (Q) or L = lim(x,y)→(x0 ,y0)

f (x , y)

Remarks:

◮ L = ∞: close to ∞ ⇐⇒ large;

◮ If such an L exists, it is unique.

ExampleHow do we prove that

lim(x,y)→(0,0)

x2y

x2 + y2= 0 ?

ExampleHow do we prove that

lim(x,y)→(0,0)

x2y

x2 + y2= 0 ?

Can’t substitute:

◮ f is not defined at P0(0, 0);

◮ Even if it were, the actual value might be different from the limit(expected value)

ExampleHow do we prove that

lim(x,y)→(0,0)

x2y

x2 + y2= 0 ?

Can’t substitute:

◮ f is not defined at P0(0, 0);

◮ Even if it were, the actual value might be different from the limit(expected value)

Polar coordinates to the rescue! x = r cos θ, y = r sin θ and

(x , y) → (0, 0) ⇐⇒ r → 0

x2y

x2 + y2=

r2 cos θr sin θ

r2= r cos2 θ sin θ

lim(x,y)→(0,0)

x2y

x2 + y2= lim

r→0r cos2 θ sin θ = 0

ExampleHow do we prove that

lim(x,y)→(0,0)

x2y

x2 + y2= 0 ?

Can’t substitute:

◮ f is not defined at P0(0, 0);

◮ Even if it were, the actual value might be different from the limit(expected value)

Polar coordinates to the rescue! x = r cos θ, y = r sin θ and

(x , y) → (0, 0) ⇐⇒ r → 0

x2y

x2 + y2=

r2 cos θr sin θ

r2= r cos2 θ sin θ

lim(x,y)→(0,0)

x2y

x2 + y2= lim

r→0r cos2 θ sin θ = 0

(Squeeze Theorem in action!)

Limits don’t always exist!

f (x , y) =xy

x2 + y2=⇒ lim

(x ,y)→(0,0)

xy

x2 + y2=?

Limits don’t always exist!

f (x , y) =xy

x2 + y2=⇒ lim

(x ,y)→(0,0)

xy

x2 + y2=?

Try the same (polar coordinates):

xy

x2 + y2= cos θ sin θ

Depends on θ

Limits don’t always exist!

f (x , y) =xy

x2 + y2=⇒ lim

(x ,y)→(0,0)

xy

x2 + y2=?

Try the same (polar coordinates):

xy

x2 + y2= cos θ sin θ

Depends on θ =⇒ trouble!

Limits don’t always exist!

f (x , y) =xy

x2 + y2=⇒ lim

(x ,y)→(0,0)

xy

x2 + y2=?

Try the same (polar coordinates):

xy

x2 + y2= cos θ sin θ

Depends on θ =⇒ trouble!Directional limit: u, nonzero vector

Limit along direction u =⇒ limt→0

f (r0 + tu)

Limits don’t always exist!

f (x , y) =xy

x2 + y2=⇒ lim

(x ,y)→(0,0)

xy

x2 + y2=?

Try the same (polar coordinates):

xy

x2 + y2= cos θ sin θ

Depends on θ =⇒ trouble!Directional limit: u, nonzero vector

Limit along direction u =⇒ limt→0

f (r0 + tu)

Example: u = 〈1,m〉 =⇒ x = t, y = mt =⇒ y = mx

limt→0

f (tu) = limx→0

mx2

x2 + m2x2=

m

1 + m2

depends on m (hence on u).

Side Limits and Directional Limits

Similarity:

◮ Single variable functions:Limit exists =⇒ side limits exist, have the same value.Conclusion: Side limits are different =⇒ limit does not exist.

◮ Multivariable functions:Limit exists =⇒ directional limits exist, have the same value.Conclusion: Directional limits have different values =⇒ limitdoes not exist.

Side Limits and Directional Limits

Similarity:

◮ Single variable functions:Limit exists =⇒ side limits exist, have the same value.Conclusion: Side limits are different =⇒ limit does not exist.

◮ Multivariable functions:Limit exists =⇒ directional limits exist, have the same value.Conclusion: Directional limits have different values =⇒ limitdoes not exist.

Difference:

◮ Single variable functions:Side limits are equal =⇒ limit exists.

◮ Multivariable functions:All directional limits have the same value =⇒ limit does notnecessarily exist.

A trickier example

lim(x ,y)→(0,0)

xy2

x2 + y4

Then along y = mx :

xy2

x2 + y4=

mx3

x2 + m4x4=

mx

1 + m2x4→ 0 as x → 0

for all m.

A trickier example

lim(x ,y)→(0,0)

xy2

x2 + y4

Then along y = mx :

xy2

x2 + y4=

mx3

x2 + m4x4=

mx

1 + m2x4→ 0 as x → 0

for all m.However, along x = y2:

xy2

x2 + y4=

y4

y4 + y4=

1

2→

1

2as x → 0

A trickier example

lim(x ,y)→(0,0)

xy2

x2 + y4

Then along y = mx :

xy2

x2 + y4=

mx3

x2 + m4x4=

mx

1 + m2x4→ 0 as x → 0

for all m.However, along x = y2:

xy2

x2 + y4=

y4

y4 + y4=

1

2→

1

2as x → 0

Conclusion:

lim(x ,y)→(0,0)

xy2

x2 + y4does not exist

Limits along paths

f : D → R , r : I → R2

such that

◮ 0 is in I , r(0) = r0;

◮ r is continuous at 0;

◮ r(t) is in D for t 6= 0

Limit along r =⇒ limt→0

f (r(t))

Limits along paths

f : D → R , r : I → R2

such that

◮ 0 is in I , r(0) = r0;

◮ r is continuous at 0;

◮ r(t) is in D for t 6= 0

Limit along r =⇒ limt→0

f (r(t))

Particular case: r(t) = r0 + tu (Directional limit)

Limits along paths

f : D → R , r : I → R2

such that

◮ 0 is in I , r(0) = r0;

◮ r is continuous at 0;

◮ r(t) is in D for t 6= 0

Limit along r =⇒ limt→0

f (r(t))

Particular case: r(t) = r0 + tu (Directional limit)Then:

limQ→P0

f (Q) = L ⇐⇒ limt→0

f (r(t)) = L

for all continuous paths r such that r(0) = r0.

Continuity

◮ f is continuous at (x0, y0) if

lim(x ,y)→(x0,y0)

f (x , y) = f (x0, y0) .

◮ f is a continuous function if it is continuous at all pointswhere it is defined.

Continuity

◮ f is continuous at (x0, y0) if

lim(x ,y)→(x0,y0)

f (x , y) = f (x0, y0) .

◮ f is a continuous function if it is continuous at all pointswhere it is defined.

Examples:

◮ Polynomial functions are continuous;

◮ Sum, difference, product, quotient of continuous functions arecontinuous;

◮ Powers, exponentials of continuous functions are continuous;

◮ Compositions of continuous functions are continuous.

Continuity

◮ f is continuous at (x0, y0) if

lim(x ,y)→(x0,y0)

f (x , y) = f (x0, y0) .

◮ f is a continuous function if it is continuous at all pointswhere it is defined.

Examples:

◮ Polynomial functions are continuous;

◮ Sum, difference, product, quotient of continuous functions arecontinuous;

◮ Powers, exponentials of continuous functions are continuous;

◮ Compositions of continuous functions are continuous.

What can go wrong:

◮ Limit different from actual value: Removable discontinuity:

◮ Limit does not exist: essential discontinuity.

Continuity of vector fields

Vector field:

F : D → R2,F(x , y) = F1(x , y)i + F2(x , y)j

F1,F2 : D → R =⇒ scalar output.

F continuous ⇐⇒ F1,F2 continuous

Continuity of vector fields

Vector field:

F : D → R2,F(x , y) = F1(x , y)i + F2(x , y)j

F1,F2 : D → R =⇒ scalar output.

F continuous ⇐⇒ F1,F2 continuous

Example:F(x , y) = y i − x j

F1(x , y) = y , F2(x , y) = −x

Polynomials =⇒ continuous components =⇒ continuous field