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Chapter 1: Problem 13

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

Short Answer

Expert verified
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

01

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'.
02

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'.
03

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'.
04

Determining the limit

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

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