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Unformatted text preview: Week 7 2-variable Limits and Continuity Summary Definitions Here is the definition of a 2-variable limit: lim ( x,y ) → ( a,b ) f ( x,y ) = L means that as ( x,y ) approaches ( a,b ) along any path, the value f ( x,y ) approaches L . If different paths give different L , we say that the limit does not exist (DNE). This is similar to the definition of a 1-variable limit lim x → a f ( x ), with the distinction that in 2 dimensions there are many different ways for ( x,y ) to approach ( a,b ) (ie different ’paths’), whereas in 1 dimension x can only approach a from the left or the right. This will matter when we want to prove non-existence (see the 2-path test below), though not really when we’re computing the value of a limit that does exist. We say that a function f ( x,y ) is continuous at ( a,b ) if lim ( x,y ) → ( a,b ) f ( x,y ) = f ( a,b ) . In other words, a function is continuous at ( a,b ) if we can compute lim ( x,y ) → ( a,b ) f ( x,y ) by just plugging in. So we can show that a function is continuous by computing a limit, but once we know that most functions are continuous, that also lets us compute many limits by plugging in. Fact: All combinations (by addition, multiplication, division, composition) of basic functions (rational functions, roots, exponentials, trigonometric, etc.) are continuous, unless there somehow is: division by zero, ln(0), or a piecewise defined function....
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