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'.