Question

*(The bottom part is x as it approaches infinity)*

I have no clue as to how you solve a limit problem when a variable is the exponent. Help pleaseee

Answer 1

`\bf \lim\limits_(x\to \infty)~\left( \cfrac{1}{8} \right)^x\implies \lim\limits_(x\to \infty)~\cfrac{1^x}{8^x}\\\\[-0.35em] ~\dotfill\\\\ \stackrel{x = 10}{\cfrac{1^(10)}{8^(10)}}\implies \cfrac{1}{8^(10)}~~,~~ \stackrel{x = 1000}{\cfrac{1^(1000)}{8^(1000)}}\implies \cfrac{1}{8^(1000)}~~,~~ \stackrel{x = 100000000}{\cfrac{1^(100000000)}{8^(100000000)}}\implies \cfrac{1}{8^(100000000)}~~,~~ ...`

now, if we look at the values as "x" races fast towards ∞, we can as you see above, use the values of 10, 1000, 100000000 and so on, as the value above oddly enough remains at 1, it could have been smaller but it's constantly 1 in this case, the value at the bottom is ever becoming a larger and larger denominator.

let's recall that the larger the denominator, the smaller the fraction, so the expression is ever going towards a tiny and tinier and really tinier fraction, a fraction that is ever approaching 0.