Deep Roots...:)

Question

Why is it that powers with very small fractional or decimal exponents all tend to one?

That is, for $x \ll 1$, $a^x \approx 1$, seemingly. True, or untrue? Can anyone offer more explanation? Thanks in advance.

Answer 1

Intuitively, it's like taking a square root with a very big index. For instance, take $100^{0.00001}=100^\frac{1}{100000}=\sqrt[100000]{100}$

This means we are looking for a number that elevated to the 100.000th power equals 100.

Notice that elevating a number $n\lt1$ to a positive exponent $x\gt1$ actually outputs a smaller number than the given one.

For this reason, the number must be bigger than $1$, but small enough that elevating it to the 100.000th power (and that is definitely a lot: $2^{100000}$ has about $10^{30100}$ digits) will only output 100.

Specularly, the same thing happens if $a\lt1$, but $a^x$ will approach $1$ from below. Since rooting $a$ actually gives a bigger output than $a$ that is smaller than $1$, increasing the index of the root will make the output tend to $1$.

Hope this example helped. Write in the comments if you need clarifications.