If a limit is infinity or negative infinity, these cannot be considered as the absolute extrema values. ... The greatest function value is the absolute maximum value and the least is the absolute minimum value.

Can an absolute maximum be infinity?

You might think that its absolute max would be infinity, but infinity is not a number and thus it doesn't qualify as a maximum (ditto for using negative infinity as an absolute min).

Can infinity be a global extrema?

There is only one global maximum (and one global minimum) but there can be more than one local maximum or minimum. Assuming this function continues downwards to left or right: The Global Maximum is about 3.7. The Global Minimum is −Infinity.