practice～ ｈｅｌｐ ～

[chuang tsai-ling]

1.Determine where the given functions are continuous. Explain clearly.

f(x)=｛ xsin(1/x) if x≠０　，　０ if x=0　｝

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2.Determine if x=0 is a point of removable discontinuity for the given functions . If it is , what should f(0) be defined to be to make f continuous at x=0?

f(x)=［sin(sin x)］/x




thanks ！:grin:

 

[HallsofIvy]

I presume you know the definitions[\b]. A function, f, is continuous at x= a if and only if \(\displaystyle \lim_{x\to a} f(x)= f(a)\). What is \(\displaystyle \lim_{x\to 0} x sin(1/x)\)?

A function has a "removable discontinuity" at x= a if and only if \(\displaystyle \lim_{x\to a}f(x)\) exists but is not equal to f(a) or f(a) is not defined. We can then "remove" the discontinuity by redefining f(a) to be equal to that limit. What is \(\displaystyle \lim_{x\to 0} \frac{sin(sin(x))}{x}\)?