Let f be a function such that lim x rightarrow a f(x) exists for any a. Define g
ID: 3079440 • Letter: L
Question
Let f be a function such that lim x rightarrow a f(x) exists for any a. Define g by g(x) = f(x + c) for some c. Then show from the epsilon - delta definition.Explanation / Answer
the mathematical statement means that for x close enough to c, the difference between f(x) and L is "small". Very similar definitions exist for functions of two or more variables; however, as you can imagine, if we have a function of two or more independent variables, some complications can arise in the computation and interpretation of limits. Once we have a notion of limits of functions of two variables we can discuss concepts such as continuity andderivatives. Limits The following definition and results can be easily generalized to functions of more than two variables. Let f be a function of two variables that is defined in some circular region around (x_0,y_0). The limit of f as x approaches (x_0,y_0) equals L if and only if for every epsilon>0 there exists a delta>0 such that f satisfies whenever the distance between (x,y) and (x_0,y_0) satisfies We will of course use the natural notation when the limit exists. The usual properties of limits hold for functions of two variables: If the following hypotheses hold: and if c is any real number, then we have the results: Linearity 1: Linearity 2: Products of functions: Quotients of functions: (provided L is non-zero) The linearity and product results can of course be generalized to any finite number of functions: The limit of a sum of functions is the sum of the limits of the functions. The limit of a product of functions is the product of the limits of the functions. It is important to remember that the limit of each individual function must exist before any of these results can be applied.
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