For $x\in R^d- \{0\}$, in dimension $d=3$, we study the asymptotic behavior of the local time $L_t^x$ of super-Brownian motion $X$ starting from $δ_0$ as $x \to 0$. Let $ψ(x)=((1/2π^2) \log (1/|x|))^{1/2}$ be a normalization, Theorem 1 implies that $(L_t^x-(1/2π|x|))/ψ(x)$ converges in distribution to a standard normal distributed random variable as $x\to 0$. For dimension $d=2$, Theorem 2 implies that $L^x_t-(1/π)\log(1/|x|)$ is $L^1$ bounded as $x\to 0$. To do this, we prove a Tanaka formula for the local time which refines a result in Barlow, Evans and Perkins.