what is the relation between limits and continuity?
Answers
Answer:
The concept of the limit is one of the most crucial things to understand in order to prepare for calculus. A limit is a number that a function approaches as the independent variable of the function approaches a given value. For example, given the function f (x) = 3x, you could say, "The limit of f (x) as x approaches 2 is 6." Symbolically, this is written f (x) = 6. In the following sections, we will more carefully define a limit, as well as give examples of limits of functions to help clarify the concept.
Continuity is another far-reaching concept in calculus. A function can either be continuous or discontinuous. One easy way to test for the continuity of a function is to see whether the graph of a function can be traced with a pen without lifting the pen from the paper. For the math that we are doing in precalculus and calculus, a conceptual definition of continuity like this one is probably sufficient, but for higher math, a more technical definition is needed. Using limits, we'll learn a better and far more precise way of defining continuity as well. With an understanding of the concepts of limits and continuity, you are ready for calculus.
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Answer:
Just as with one variable, we say a function is continuous if it equals its limit: A function f(x,y) is continuous at the point (a,b) if lim(x,y)→(a,b)f(x,y)=f(a,b). A function is continuous on a domain D if is is continuous at every point of D.