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.