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
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