The document discusses functions of several variables and concepts related to limits and continuity of such functions. It provides examples of functions of multiple variables, defines the limit of a function of two variables as the point approaches a value, and defines continuity of a function of two variables. It then gives three examples: 1) showing that a function's iterated limits exist but the simultaneous limit does not, 2) evaluating the limit of a function as the point approaches the origin, and 3) showing that a function is continuous at the origin.

1. Concepts Used
Functions of Several Variables
A real valued function of real variables x1, x2, x3......, xn is a function which associates
with each element (x1, x2, …... xn) a unique element
i.e.
where x1, x2, …... xn are n independent real variables and u is a dependent variable.
Examples :
1. u = x3 + y2 + z3 + 3xyz is a function of three di
ff
erent variables x, y, z.
2. u = x2 + y2 is a function of two variables x, y.
Limit of a Function of Two Variable
s
A function is said to be approaching limit l as the point approaches to point
(a, b) if given ∈ > 0, however small, there exists a positive real number δ (depending on ∈)
such that whenever
We write :
Note : If exists, then this limit is independent of the path along which we
approach the point (a, b)
Continuity of a Function of Two Variables
A function f(x, y) is said to be continuous at a point (a, b) if given ∈ > 0, however small,
there exists a positive real number δ (depending on ∈) such that :
whenever
Equivalently, function f(x, y) is continuous at point (a, b) if exists and
f n
f(x1, x2, … . . . xn)
u = f(x1, x2, … . . . xn)
f(x, y) (x, y)
| f(x, y) − l| < ϵ |(x, y) − (a, b)| < δ
lim
(x,y)→(0,0)
f(x, y) = l
lim
(x,y)→(a,b)
| f(x, y) − f(a, b)| < ϵ |(x, y) − (a, b)| < δ
lim
(x,y)→(a,b)
f(x, y)
lim
(x,y)→(a,b)
f(x, y) = f(a, b) .
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2. Examples
1. If a function is de
fi
ned by = , (x, y) (0,0)
Show that the two iterated limits : and exist but
the simultaneous limit does not exist.
Sol. = = = 1
= = = -1
The two iterated limits exist.
Let along the line , then
= =
=
which is not unique as it takes di
ff
erent values for di
ff
erent values of m.
The simultaneous limit does not exist.
f f(x, y)
x2
− y2
x2 + y2
≠
lim
x→0
[lim
y→0
f(x, y)] lim
y→0
[lim
x→0
f(x, y)]
lim
(x,y)→(0,0)
f(x, y)
lim
x→0
[lim
y→0
f(x, y)] lim
x→0
[lim
y→0
x2
− y2
x2 + y2
] lim
x→0
x2
x2
lim
y→0
[lim
x→0
f(x, y)] lim
y→0
[lim
x→0
x2
− y2
x2 + y2
] lim
y→0
−y2
y2
∴
(x, y) → (0,0) y = mx
lim
(x,y)→(0,0)
f(x, y) lim
(x,y)→(0,0)
x2
− y2
x2 + y2
lim
x→0
x2
− m2
x2
x2 + m2x2
lim
x→0
1 − m2
1 + m2
∴ lim
(x,y)→(0,0)
x2
− y2
x2 + y2
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3. 2. Show that = 0
Sol. = Put ,
=
=
=
=
=
lim
(x,y)→(0,0)
x4
− y4
x2 + y2
|
x4 − y4
x2 + y2
− 0| |x2
− y2
| x = r cosθ y = r sinθ
|r2
cos2
θ − r2
sin2
θ|
|r2
(cos2
θ − sin2
θ)|
|r2
(cos2θ)|
r2
|(cos2θ)| ≤ r2
[ ∵ |cos2θ| ≤ 1]
x2
+ y2
< ϵ if x2
<
ϵ
2
and y2
<
ϵ
2
, where ϵ > 0
∴ |
x4
− y4
x2 + y2
− 0| < ϵ if |x|2
<
ϵ
2
and |y|2
<
ϵ
2
or |
x4
− y4
x2 + y2
− 0| < ϵ if |x| <
ϵ
2
and |y| <
ϵ
2
or |
x4
− y4
x2 + y2
− 0| < ϵ if |x| < δ and |y| < δ where δ =
ϵ
2
or |
x4
− y4
x2 + y2
− 0| < ϵ if |x − 0| < δ and |y − 0| < δ
∴ lim
(x,y)→(0,0)
x4
− y4
x2 + y2
= 0
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4.
3. Show that the function is continuous at origin.
Sol. Let , however small
Then
is continuous at origin.
f(x, y) = |x| + |y|
ϵ > 0
| f(x, y) − 0| = ||x| + |y| − 0|
= ||x| + |y||
≤ |x| + |y|
<
ϵ
2
+
ϵ
2
if |x| <
ϵ
2
and |y| <
ϵ
2
∴ | f(x, y) − 0| < ϵ if |x − 0| < δ and |y − 0| < δ, where δ =
ϵ
2
∴ f(x, y)
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