Math 211- Yahdi

Section 14.2: Find the limit of f(x,y) as (x,y) approaches (0,0), if it exists, or show that it does not exist.

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Key Points:

1-  Concept/Definition/Laws of the limit for 2 variables = similar to one variable function

2-  (x,y) can approach (a,b) in infinitely many paths. (useful to prove the limit DNE):

3-  Limit Graphically at (0,0): Guess the limit or it existence for f1, f2, …, f6 using the graph surfaces? à Works for f1 & f2, but not clear for f3 to f6

4-  Limit at (0,0) using contour map: Guess the limit or its existence for f1,f2,..,f6 using the contour maps? Note that each color shade corresponds to a different value of the output z=f(x,y). à Can guess the existence for f1,f2,f3 & the non-existence for f4,f5,f6

5-  Limit at (0,0) by substitution: If substitution works using the arithmetic of limits, you are done. Practice with f1, f2 & f3

6-  Limit at (0,0) when substitution:

  1. The limit does not exist: It is more likely the case when the degree of the numerator is smaller than or equal to the degree of the denominator.
  1. You need to find two different paths for which the limits are different. You can guess the paths by looking at the contour map and/or using the following paths: Along the x-axis, the y-axis, the line x=y, the lines y=mx, the curves y=x^n or the curves x=y^n.
  2. The last two paths are determined to be able to transform the denominator to term with the same powers (like for f6)
  3. Practice with f4, f5, f6
  1. The limit does exist: In general this is more likely the case when the degree of the numerator is strictly larger than the degree of the denominator while substitution does not work! There are different method to find the limit then:
  1. Factor out and simplify: Practice with f9
  2. Substitute a repeated expression of x & y by u then using limit of one variable techniques: Practice with fq0

iii.  Squeeze theorem (also valid for 2-variables function): Practice with f11.

7-  Limit at any (a,b): Works the same as the limit at (0,0).