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Comment: Multivariate limits
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| One way to express the limit of a multivariate function is to imagine a disk around a target point. The disk is principally formed by all input coordinates no more than ''δ'' away from the target coordinate (leaving out the target itself). This set of input coordinates map to a set of outputs characterized by ''|f(·) - L| < ε'', i.e. these outputs are at most ''ε'' away from the limit. As ''δ'' shrinks to 0, so too does the set of input coordinates, and so too does ''ε''. | One way to express the limit of a multivariate function is to imagine a disk (or ball) around a target point. The disk (ball) is principally formed by all input coordinates no more than ''δ'' away from the target coordinate (leaving out the target itself). This set of input coordinates map to a set of outputs characterized by ''|f(·) - L| < ε'', i.e. these outputs are at most ''ε'' away from the limit. As ''δ'' shrinks to 0, so too does the set of input coordinates, and so too does ''ε''. |
Limit
A limit is the value that a function approaches as the input approaches some value.
Description
The limit of f as x approaches infinity is notated as:
![[ATTACH]](/Calculus/Limit?action=AttachFile&do=upload_form&target=lim1.svg&ticket=0069e00c61.d5c6daab81d3dccb3eff125b594cc1288880dfc4 "[ATTACH]"){kind=small}
Vector-Values Function
The limit of the vector-values function r(t) = f(t)i + g(t)j + h(t)k as t approaches a is given by:
provided that the component limits exist.
Multivariate Function
One way to express the limit of a multivariate function is to imagine a disk (or ball) around a target point. The disk (ball) is principally formed by all input coordinates no more than δ away from the target coordinate (leaving out the target itself). This set of input coordinates map to a set of outputs characterized by |f(·) - L| < ε, i.e. these outputs are at most ε away from the limit. As δ shrinks to 0, so too does the set of input coordinates, and so too does ε.
Consider the limit of f(x,y) as it approaches (a,b). There is a disk of (x,y) coordinates around (a,b) such that:
The limit is found by shrinking δ. This is notated as:
