Part 2/6:
However, the expression ( 0^0 ) introduces complexity. While some contexts might define ( 0^0 ) as one, it is typically considered undefined in mathematical terms due to the ambiguity that arises when trying to interpret 0 raised to any power.
Analyzing Exponential Functions
To better grasp these ideas, let us plot a simple exponential function like ( 3^x ). Observing its behavior, we note crucial points:
As ( x ) approaches negative infinity, ( 3^x ) approaches zero. This means ( \lim_{x \to -\infty} 3^x = 0 ).
At ( x = 0 ), ( 3^0 ) equals 1, maintaining our earlier finding.
As ( x ) approaches positive infinity, ( 3^x ) grows without bound, heading toward infinity.