1) Isn't the denominator of the infinite limit just unity?

2) This is why we have Rules of Thumb. Here are a couple:

Use the t-distribution until the sample size is "big enough".

The Normal is a good approximation of Binomial if n is big enough and p is close to 1/2.

3) I'm not convinced that the mean is a good measure of t being big enough. Outliers are bad!

My views. I welcome others'.