# What happens when something grows exponentially?

Oct 31, 2014

In general: For an exponential function whose exponent tends to $\pm \infty$ as $x \to \infty$, the function tends to $\infty$ or 0 respectively as $x \to \infty$.

Note that this applies similarly for $x \to - \infty$ Further, as the exponent approaches $\pm \infty$, minute changes in $x$ will (typically) lead to drastic changes in the value of the function.

Note that behavior changes for functions where the base of the exponential function, i.e. the $a$ in $f \left(x\right) = {a}^{x}$, is such that $- 1 \le a \le 1$.

Those involving $- 1 \le a < 0$ will behave oddly (as the $f \left(x\right)$ will not take on any real values, save where $x$ is an integer), while ${0}^{x}$ is always 0 and ${1}^{x}$ is always 1.

For those values $0 < a < 1$, however, the behavior is the opposite of the long-term behavior noted above.

For functions ${a}^{x}$ with $0 < a < 1$, as $x \to \infty$, $f \left(x\right) \to 0$, and as $x \to - \infty$, $f \left(x\right) \to \infty$