At the point $P\left(a,a^n\right)$ on the graph of $y=x^n,\left(n\in N\right)$ , in the first quadrant, a normal is drawn. The normal intersects the $y-a\xi s$ at the point $\left(0,b\right)$ . If ${\left(\text{lim}\right)}_{a\vec{0}}=\frac{1}{2},$ then $n$ equals _____.

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