** Next:** Continuous functions.
** Up:** Multivariable functions.
** Previous:** Level curves.

**Definition 4.4**
Let

*f* be a function defined on a (pointed) neighborhood of

*P*_{0}=(

*x*_{0},

*y*_{0}) and

.
The function

*f* has a limit equal to

*l* when

*P*=(

*x*,

*y*) is arbitrarily close to

*P*_{0} if:

We denote:

**Proposition 4.5**
If it exists, the limit of *f* at *P*_{0} is unique.

**Example 4.6**
Let

*f* be the function defined by

Approach the origin on the parabola

whose equation is

*y*=

*mx*^{2}, where

*m* is a real parameter
different from 0. On

,
we have:

Thus the limit at 0 of the function

*f*, computed on the parabola

is equal to

,
i.e. on each such parabola, there is a separate ``candidate'' to be the limit of

*f* at (0,0). Thus there is no limit for

*f* at the origin.

**Proposition 4.7** (Algebra of the limits)
Let

*f* and

*g* be two functions, and suppose that

Then:

- 1.
__Sum:__
.
- 2.
__Product:__
.
- 3.
__Constant multiple:__
.
- 4.
__Quotient:__ If
.
- 5.
__Powers:__ If
is a rational number and if *l*_{1} >0, then
.

**Example 4.8**

- Let
*f*(*x*,*y*)=*x*^{2}*y*+2*xy*^{3}-1. Then:

- Let
.
Then:

- Let
.
Then:

**Example 4.9**
Compute the limit at (0,0) of the function

*f* such that

Both the numerator and the denominator approach 0, when (

*x*,

*y*) approaches the origin, thus we cannot use the
quotient rule of prop.

4.7. We multilply the numerator and the deniminator by the conjugate
expression of the denominator; we have:

Hence:

The main usage of that proposition is in disproving the existence of a limit.

**Example 4.11**
Let

define a function on the whole plane, but not at the origin.

We will approach the origin on a line, whose equation is *y*=*mx*. Then:

If the values

*m*_{1} and

*m*_{2} of the parameter

*m* verify the condition

,
the limits of

*f*(

*x*,

*mx*) at 0 are diferent. Therefore, the function

*f* has no limit at the origin.

** Next:** Continuous functions.
** Up:** Multivariable functions.
** Previous:** Level curves.
*Noah Dana-Picard*

*2001-05-30*
n