Asymptotic equivalence using binomial series

Question

I want to prove that $ t^3-1-3(t-1) \sim 3(t-1)^2 $ as $ t \to 1 $ using binomial series. Any advice?

Answer 1

Hint: By binomial theorem, writing $t=1+(t-1),$ we get:

$$t^3=\left(1+(t-1)\right)^3=1+3(t-1)+3(t-1)^2+(t-1)^3.$$

So $1+3(t-1)$ are the first terms of $t^3$ near $t=1.$ The right side, $3(t-1)^2$ is the next term.

Answer 2

Using the binomial theorem, $(t-1)^3=t^3-3t^2+3t-1.$

As $t\to1$, $(t-1)^3\to0$, so $t^3-3t^2+3t-1\to0,$

so $t^3-1-3(t-1)=t^3-3t+2\sim3t^2-6t+3=3(t-1)^2.$