Finding limit- greatest integer function

[life is maths]

Hi, I was studying for my upcoming calculus exam and couldn't be sure if I could solve this question:

$\stackrel{lim}{x\rightarrow0}$ x [$\frac{1}{x}$ ]

If x approaches 0 from left, then 0< x [$\frac{1}{x}$ ]<1

If x approaches 0 from right, then x [$\frac{1}{x}$ ]>1 since x [$\frac{1}{x}$ ]=$\frac{x}{x-1}$

Hence, the limit is equal to 1.

Is this solution true? Thank you for any help :)

 

[1994Bhaskar]

we know that t-1< [t] <= t.
Replace here t by 1/x.
I think your answer is correct.This form of inequality is more formal method.

 

[life is maths]

Thanks, Bhaskar. You're right, I should be able to write it formally.