Difference between these two limits I want to know the difference between the following limits:
$$\lim_{x\to 1^{+}} \frac{1}{\lfloor{x}\rfloor-1},$$
$$\lim_{x\to 1^{+}} \frac{1}{x-1}$$
Is the answer to both infinite?
What is the difference between undefined and infinite?

Short answer: Some calculus textbooks say that an infinite limit is undefined, so there is no difference between the two, while others say infinite limits are defined, so there is a difference.

Longer answer:
In the first limit, if $x$ is close to $1$ but to the right of it  then $\lfloor x\rfloor=1$, so the expression inside the limit is $\frac 10$ and is undefined. That means the limit is completely undefined.
In the second limit, the expression inside the limit is defined for $x$ close to $1$ but to the right of it. However, as $x\to1^+$, the expression increases without bound.
There is therefore no real number that is the second limit. Some calculus textbooks say that the limit is undefined.
However, some calculus textbooks say that the limit is defined and equals $+\infty$. In other words, they agree that the limit is not a real number, but they add two more "numbers" so some more limits exist. The real numbers plus the "numbers" $+\infty$ and $-\infty$ are called the "affine extended real number system." Looking at limits this way has the advantage that you can talk about differences between the two kinds of limit you show. There is also the disadvantage that some theorems that are true with finite limits are not true if you allow infinite limits, so the theorems must be more carefully stated.
I have seen and used both kinds of calculus textbooks, and both are common. I cannot recommend one kind over the other, in general. The two philosophies do sometimes make some questions ambiguous, as in your question. The calculus textbook I now use to teach sticks infinite limits into one skippable section, so outside that section it treats infinite limits as undefined. In another section it does explain multiple causes for limits to be undefined and would distinguish your two kinds of causes (as well as others such as oscillations).
