Keeping it simple. Once you've got the mantissa and exponent out, you can check your number is in range by a simple check that the exponent is within range.

For a 32 bit signed integer the limit is 2E9. This means that the exponent is fine 0-8, or if the exponent is 9, then only if the mantissa is 1 or 2. This only works with a single digit mantissa.

For adding more digits to the mantissa, while a robust range check can be done, it gets more complicated. String-to-Integer functions are very conservatively written pieces of code designed to be quick.