\[\lim_{x \rightarrow 0}\left| \left| x \right|^{\beta}\times \frac{\left| 1-e ^{-x} \right|}{\left| x \right|} \times sin(\frac{1}{x})\right|\]\[\beta>1\]

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\[\lim_{x \rightarrow 0}\left| \left| x \right|^{\beta}\times \frac{\left| 1-e ^{-x} \right|}{\left| x \right|} \times sin(\frac{1}{x})\right|\]\[\beta>1\]

Mathematics
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hmm I'm thinking what if we divide by that abs(x) on the bottom
so the lim becomes |x|^(beta-1)
hmm interesting problem, let me get over some of my own homework, then will focus on this. Maybe someone else can help meanwhile.

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