Why does the graph of an exponential function shoot straight up when getting to x=1 in an exponential growth function with x^huge number?

Question

I used to notice that when x is raised to the power of a huge number, the graph shoots up at x=1. Why does this happen?

Because $1^{\text{huge number}}=1$ but say $1.01^{\text{huge number}}\ge1+0.01\times\text{huge number}$. So if $\text{huge number}>10000$ then already the graph is higher than $100$ for just $x=1.01$. — , May 02 '15 at 22:40

score 0 · Answer 1 · answered May 02 '15 at 22:12

Intuitively, if $f(x) = x^a$ with $a$ a very large positive number, then if $x$ is just larger than $1$, $x^a$ will also be very large. For example, $1.001^{10^6}$ is about $10^{434}$.

Using calculus, $f'(x) = ax^{a-1}$, so for $x$ just larger than $1$, this will be about $a$, so the slope of $f$ is very large to the right of $x=1$.

Why does the graph of an exponential function shoot straight up when getting to x=1 in an exponential growth function with x^huge number?

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