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Continuous function

Yankel

Active member
Jan 27, 2012
398
Hello

I need some help with this question please:

For which values of a the next function is continuous at x=0 ?

[tex]\left\{\begin{matrix} x^{a}\cdot sin\frac{1}{x} & x\neq 0\\ 0 & x=0 \end{matrix}\right.[/tex]

I know that for it to be continuous at x=0, I need f(0)=lim x-->0

So I tried calculating the limit, and got to:

[tex]\lim_{x\to0}x^{a-1}\cdot sin\frac{1}{x}\cdot x[/tex]


not sure I am correct, but anyhow do not know how to proceed. What I tried to do was to bring the limit to a known form
 

Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,192
I would think about the squeeze theorem, and the fact that $-1\le \sin(x)\le 1\;\forall\,x$.