Does the limit $\lim_{x \to 1}\left(\frac{x}{\lfloor x\rfloor}\right)$ exist? Where $\lfloor x\rfloor$ is the Greatest Integer Function or the Floor function.
My teacher defined that a limit exists when the Left-hand Limit = Right-Hand Limit and If any one is undefined but the other is finite then also the limit exists. Like $\lim_{x \to 0}\sqrt{x}$ does exist although the LHL does not lie in the domain.
In this question the domain of the function $\frac{x}{\lfloor x\rfloor}$ is $\mathbb R\setminus[0,1)$ where $\mathbb R$ represents Real Numbers. So, The Left-Hand Limit is not defined and Right Hand Limit is $1$, which is finite so according to my teacher Limit does exist. But when I tried this question in WolframAlpha, it is saying that the LHL is $-\infty$ and RHL is $1$, so limit does not exist.
So, a more general question is: do we have to consider the domain of a function when we calculate a limit?
Best Answer
Your teacher is completely right, according to the more general definition of limit, we consider $x$ in the domain and therefore
$$\lim_{x \to 1}\left(\frac{x}{[x]}\right)= \lim_{x \to 1^+}\left(\frac{x}{[x]}\right)=1$$
Refer also to the related