Limits of two variables $\lim_{t \rightarrow \infty} \lim_{u \rightarrow t} u^{\alpha} \sqrt{t-u} = 0$

Question

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I want to find $\alpha>0$ to satisfy

$$ \lim_{t \rightarrow \infty} \lim_{u \rightarrow t} u^{\alpha} \sqrt{t-u} = 0 $$

My intuitive guess is $\alpha$ must be less than $1/2$. But how to prove this rigorously?

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Edit - I asked a sort of a dumb question.. I will post the original question.