Question

Find the limit (if it exists). If it does not exist, explain why. $$ \lim _{x \rightarrow 3}(2-\llbracket-x \rrbracket) $$

Short answer

The limit of the function as x approaches 3 does not exist because the limits from the left side and the right side do not converge to the same value.

Step-by-step solution

Understand the floor function

The floor function, \(\llbracket -x \rrbracket\) , represents the greatest integer less than or equal to '-x'. It rounds '-x' down to the nearest integer; therefore, we can write \(\llbracket -x \rrbracket = -n\) , where 'n' is the smallest non-negative integer that is still greater than 'x'.

Substitute and find limit from the left side

As 'x' approaches '3' from the left, '-x' becomes slightly less than '-3', hence the floor function \(\llbracket -x \rrbracket\) would yield '-4'. Substitute '-4' into the function, yielding \((2 - (-4)) = 6\). Thus, the limit from the left side is '6'.

Substitute and find limit from the right side

As 'x' approaches '3' from the right, '-x' becomes slightly larger than '-3'. Hence the floor function, \(\llbracket -x \rrbracket\), gives '-3'. Substitute '-3' into the function, yielding \((2 - (-3)) = 5\). Therefore, the limit from the right side is '5'.

Determining the limit

Because the limit from the left side and the right side do not converge to the same value (6 ≠ 5), we conclude that the limit of the function does not exist at 'x=3'.