Question

How do you find the limit of `(((2+x)^3) -8 )/ x` as x approaches 0?

Answer 1

12

Explanation:

`lim_{x to 0} ((2+x)^3 -8 )/ x`

if we start the binomial expansion of the cubic in the numerator [just use Pascal's Triangle] to Order `x^2` we have

`lim_{x to 0} ((8 + 3(4) x + mathcal{O} (x^2)) -8 )/ x`

`= lim_{x to 0} (12 x + mathcal{O} (x^2)) / x`

`= lim_{x to 0} 12 + mathcal{O} (x) = 12`