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In a Nut Shell:
Suppose
f(x,y) approaches the value of L as (x,y) approaches the value of
(a,b). Then the limit means that the distance
between f(x,y) and L can be made
arbitrarily
small by making the distance
between (x,y) and (a,b) sufficiently small (but not zero).
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The formal definition of the limit:
lim f(x,y) = L
(x,y) →(a,b)
If for every number ε > 0
there is a corresponding number
δ > 0
such that
if (x,y) ϵ D and
0 < √[ (x ˗ a)2 +
(y ˗b)2 ] < δ
then | f(x,y) ˗ L < ε .
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Strategy for finding limits:
It's easiest for cases where
the limit does not exist. So first
attempt to show that the limit
does not exist. In this case, the strategy is to approach
the limit of the function from
different
directions. If any two directions
yield a different limit, then the limit of the
function does not exist.
If you try several
directions and each yields the same limit, then perhaps the limit does
exist. This type is much more difficult. Strategies to show the limit exists and
find the
value of the limit in
this case include:
a. If the function is a polynomial or a
rational function then use continuity to find
its limit.
b. Use the squeeze theorem to determine the
limit.
c. Use the formal definition of the limit
directly. (as given above)
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Squeeze Theorem for functions with two
independent variables:
If f(x,y)
≤ g(x,y) ≤
h(x,y) when (x,y) is near (a,b)
except possibly at (a,b) and
lim f(x,y) = lim h(x,y)
= L
(x,y) → (a,b) (x,y) → (a,b)
then lim g(x,y)
= L
(x,y) → (a,b)
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