Limit of the function$f(x) = \frac{x-1}{x-x^a}$ in 1

Question

I am struggling with the limit of this function :

$$f(x) = \frac{x-1}{x-x^a} $$

as $x$ tends to 1 and $x>1$. The constant $a$ is lower than 1, i.e., $a\in] \infty, 1[$.

My guess is that the limit $\lim_{x\rightarrow 1, x>1} f(x) $ exists and is finite.

Could someone proove this (exact limit or upper bound).

Thank you very much !

Answer 1

Since the denominater tends to 0 as $x$ goes to 1 we use L’hôpital’s rule for finding limits. Using L’hôpital’s rule we get $$\lim_{x\to 1}\frac{x-1}{x-x^a}= \lim_{x\to 1}\frac{1}{1-ax^{a-1}}= \frac{1}{1-a}$$ Note that since 1 raised to any power is 1, $\lim_{x \to 1}x^{a-1}=1$.