Limit Considerations w/o L'Hôpital on a Quotient of Root Expressions

[tellmesomething]

Homework Statement

$$\lim_{x\to 1} \frac{\sqrt[n]{x}-1}{\sqrt[m]{x}-1}$$ ($m$ and $n$ are integers)

Relevant Equations

None

I know how to do this using lopital since its a 0/0 indeterminate form. However I would like to do it without using lopital as well..how should I go about it? For starters I thought of rationalizing the numerator and denominator but we cant necessarily apply the (a+b)(a-b) identity since we dont know if m and n are odd or even integers.. Please provide a hint..

 

[Euge]

Observe $$\lim_{x\to 1} \frac{\sqrt[n]{x}-1}{\sqrt[m]{x}-1} = \lim_{x\to 1} \frac{\sqrt[n]{x} -1}{x-1} \cdot \lim_{x\to 1} \frac{x-1}{\sqrt[m]{x}-1}$$ Use the identity $$t^n - 1 = (t-1)(t^{n-1} + t^{n-2} + \cdots + t + 1)$$ with $t = \sqrt[n]{x}$ to simplify $\lim_{x\to 1} \frac{\sqrt[n]{x}-1}{x-1}$. Use a similar factorization to compute $\lim_{x\to 1} \frac{x-1}{\sqrt[m]{x}-1}$.

 

[tellmesomething]

$$t^n - 1 = (t-1)(t^{n-1} + t^{n-2} + \cdots + t + 1)$$

Thankyou! I was looking for this general form.