define limit of a function of two real variables
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For a two variable function f(x, y) the limit L can be defined as,
lim(x,y)→(a,b)f(x,y)=Llim(x,y)→(a,b)f(x,y)=L
The distance from (a, b) to L will be really small, that is, δδ but will not be zero.
Limit of a Function of Two Variables
For the function f(x, y), the limit L at (a, b) is defined when 0<(x−a)2+(y−b)2−−−−−−−−−−−−−−−√<δ0<(x−a)2+(y−b)2<δ when |f(x,y)−L|<ϵ|f(x,y)−L|<ϵ .
Limit of a two variable function will have the given characteristics.
1. The limit should be unique, if it exists.
2. The limit of sum of two functions, difference of two functions and product of two functions is the sum, difference and product of the limits.
3. Limit of quotients is the quotient of the limits given the denominator is non-zero.
How to find limit of a function
To get the limit of the function lim(x,y)→(a,b)f(x,y)lim(x,y)→(a,b)f(x,y) we need to follow the given steps:
1. Apply the limit for the value of x = a and get the value.
2. If limit is given to be infinity, change the value of function such that it comes in terms of 1x1x and then apply the limits.
3. Again, apply the limit y = b and get the value.
4. If both values come same, the limit exists and it is the obtained value.
5. L'hospital rule is not applied on functions with two variables
Example 1: Find the limit of the function f(x,y)f(x,y) = x2x2+y3x2x2+y3 at (0, 0).
Solution: First we will check if the limit exists at (0, 0) for f(x, y).
f(0,y)f(0,y) = 00+y300+y3 = 0y30y3 = 00
f(x,0)f(x,0) = x2x2+0x2x2+0 = 11
Hence, limit does not exist
lim(x,y)→(a,b)f(x,y)=Llim(x,y)→(a,b)f(x,y)=L
The distance from (a, b) to L will be really small, that is, δδ but will not be zero.
Limit of a Function of Two Variables
For the function f(x, y), the limit L at (a, b) is defined when 0<(x−a)2+(y−b)2−−−−−−−−−−−−−−−√<δ0<(x−a)2+(y−b)2<δ when |f(x,y)−L|<ϵ|f(x,y)−L|<ϵ .
Limit of a two variable function will have the given characteristics.
1. The limit should be unique, if it exists.
2. The limit of sum of two functions, difference of two functions and product of two functions is the sum, difference and product of the limits.
3. Limit of quotients is the quotient of the limits given the denominator is non-zero.
How to find limit of a function
To get the limit of the function lim(x,y)→(a,b)f(x,y)lim(x,y)→(a,b)f(x,y) we need to follow the given steps:
1. Apply the limit for the value of x = a and get the value.
2. If limit is given to be infinity, change the value of function such that it comes in terms of 1x1x and then apply the limits.
3. Again, apply the limit y = b and get the value.
4. If both values come same, the limit exists and it is the obtained value.
5. L'hospital rule is not applied on functions with two variables
Example 1: Find the limit of the function f(x,y)f(x,y) = x2x2+y3x2x2+y3 at (0, 0).
Solution: First we will check if the limit exists at (0, 0) for f(x, y).
f(0,y)f(0,y) = 00+y300+y3 = 0y30y3 = 00
f(x,0)f(x,0) = x2x2+0x2x2+0 = 11
Hence, limit does not exist
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