What happens when something grows exponentially?

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What happens when something grows exponentially?

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Answer 1 · 2014-10-31T23:30:48

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

Note that this applies similarly for $x \to - \infty$ $x->-oo$ Further, as the exponent approaches $\pm \infty$ $+-oo$ , minute changes in $x$ $x$ 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 $a$ $a$ in $f (x) = a^{x}$ $f(x) = a^x$ , is such that $- 1 \leq a \leq 1$ $-1<=a<=1$ .

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

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

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

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What happens when something grows exponentially?

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