How the z-distribution is related to the t-distribution

As the number of degrees of freedom increases, the $t$ -distribution becomes closer and closer to the $z$ -distribution.

Most tables of the $t$ -distribution have their last entry listed as either degrees of freedom =$\infty$ or $z$ , which reflects the fact that, when the number of degrees of freedom is large (say, several hundred or more), the two curves are nearly indistinguishable.

The animation below shows the bell-shaped curves of the $z$ distribution (in red) and the $t$ distribution (in green) superimposed on one another.

The $t$ -distribution curve changes shape as the number of degrees of freedom increases, and quickly becomes nearly identical to the $z$ -distribution curve.