Is $\lim_{x \to -∞} (2+3x)^{2/3}$ positive or negative?

Question

$\lim_{x \to -∞} (2+3x)^{2/3}$

Is this $(-∞)^{2/3} = (-∞^2)^{1/3} = +∞$ ?

Answer 1

It is hard to understand the number of deleted answers to this question, and the number of nonsensical comments on this page.

You are essentially asking whether $x ^{\frac 2 3}$ has a meaningful definition when $x<0$. Of course it does, maybe you'll see it more clearly if you write it as $(\sqrt[3] x)^2$ or, equivalently, $\sqrt[3] {x^2}$. Both expressions involving cubic roots do make sense. In fact, $x^{\frac p q}$ makes obvious sense when $q$ is odd, no matter who the real number $x$ is.

In particular, since $(\sqrt[3] x)^2$ is a square of a real number, it is positive, so your limit is indeed $+\infty$.

Answer 2

HINT:

Does the given limit exist finitely at all?

Answer 3

$$(2+3x)^{2/3}=x^{2/3}\left(3+\frac2x\right)^{2/3}\xrightarrow[x\to-\infty]{}+\infty\cdot3^{2/3}=+\infty$$