Why activation functions used in neural networks generally have limited range?
The main goal of an Activation function is to add non-linearity but at the same time


It must not explode the large inputs else time to reach minima will become very long
It should have a smooth gradient, this will help the training keep going towards minima without being stuck


With your example function - 

We will achieve non-linearity but gradient will be very large for big value and it will slow down the training

I believe you guessed this, so you asked the next question. What if I normalize the data - 

Normalization helps in bringing two different features at the same level. But the larger value of the same feature will still be larger compared to the smaller one

But such function stops learning at both the extreme due to flattening gradient. Also, they require high computation
We have RELU without an upper limit. It's a good trade-off on all the mentioned points