The greatest integer function is defined by $$ \lfloor x \rfloor = \text{the largest integer that is less than or equal to x}$$Let $$f(x) = \lfloor x \rfloor + \lfloor -x \rfloor$$ Show that $$\lim_{x \to 2}\; f(x)$$ exists but is not equal to f(2).

Question

The greatest integer function is defined by  $$ \lfloor x \rfloor = \text{the largest integer that is less than or equal to x}$$Let $$f(x) = \lfloor x \rfloor + \lfloor -x \rfloor$$  Show that $$\lim_{x \to 2}\; f(x)$$ exists but is not equal to f(2).

cruffo · Answer

Repost:

The greatest integer function is defined by  $$ \lfloor x \rfloor = \text{the largest integer that is less than or equal to x}$$Let $$f(x) = \lfloor x \rfloor + \lfloor -x \rfloor$$  Show that $$\lim_{x \to 2}\; f(x)$$ exists but is not equal to f(2).

watchmath · Answer

Note that $\lfloor x\rfloor +\lfloor -x\rfloor\leq x+(-x+1)=1$

anonymous · Answer

$$\lim_{x \rightarrow 2^+}f(x)=[2^+]+[-2^+]=2-3=-1$$$$\lim_{x \rightarrow 2^-}f(x)=[2^-]+[-2^-]=1-2=-1$$
f(2)=0

watchmath · Answer

dipankar is right :)

watchmath · Answer

we don't need to do in general since we are only interested for x near 2.

cruffo · Answer

That was a quick answer!