From Googling, it seems to be that the argument focuses on looking at the pdf (probability density function) of the t-distribution and the normal distribution. Certainly, you can use limits to show that the t pdf tends toward the normal pdf. But why n=30?
This page says "at values of ν [the sample size] as small as 10 or 12, the graphs of [t pdf] are nearly indistinguishable from graphs of the standard normal probability density function, and by the time ν is as large as 29 or 30, results using the t-distribution agree with results from the standard normal distribution to within a percentage point or two, and so statisticians tend to use the standard normal probability tables in place of t-tables whenever the value of ν is larger than 29 or 30."
So they use a benchmark of "a percentage point or two" difference in pdf?
At n=30, the difference in CDF between the t and normal is 0.005 (
as seen here).
So n=30 is good. Are you wondering why not n=10?