Taking the limit of the Solution of the Diffusion Equation

Question

I am currently trying to show a double limit for the convergence of the solution of the Diffusion Equation. I showed that given a PDE $u_t=u_{xx}$ and auxiliary condition $u(x,0) = g(x)$, that: $$\lim_{t\rightarrow 0} u(x,t)=g(x)$$ Where $u(x,t)$ is given by: $$u(x,t)= \int_{-\infty}^\infty \Phi(x-y,t)g(y) fy$$ And $\Phi$ is the Heat Kernel. Now, I need to show that: $$\lim_{t\rightarrow 0, y\rightarrow x}u(y)=g(x)$$ How do I go about solving this? I solved the case above when it was just $t\rightarrow 0$, but I don't know how to begin with solving a double limit. If anyone could help it would be appreciated. Thank you