  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 $\delta_0$
as $x \to 0$. Let $\psi(x)=((1/2\pi^2) \log (1/|x|))^{1/2}$ be a normalization,
Theorem 1 implies that $(L_t^x-(1/2\pi|x|))/\psi(x)$ converges in distribution
to a standard normal distributed random variable as $x\to 0$. For dimension
$d=2$, Theorem 2 implies that $L^x_t-(1/\pi)\log(1/|x|)$ 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.
