Evaluating $\lim_{x\to -\infty} \frac{(x-1)}{(x^{2/3}-1)}$

Question

The limit at negative infinity should not exist, right? $$\lim_{x\to -\infty} \frac{(x-1)}{(x^{2/3}-1)}$$ for positive infinity, the limit is infinity, but the function is undefined for values less zero. Am I going about this correctly?

Thanks

Answer 1

You are mistaken. The expression is perfectly well-defined for values of $x$ less than zero.

Specifically, since $x$ is diverging to $-\infty$, you may assume it is large in magnitude and negative. Then it is certainly beyond any zeroes of the denominator, and the rational exponent is not problematic as it is an odd root of an even power.

As you allow infinite limits, the limit is $-\infty$ because the magnitude increases without bound and the numerator is eventually negative while the denominator is eventually positive.