For any real number x, the floor function (or greatest integer function) ⌊x⌋ is the greatest integer less than or equal to x (see figure).


a. Compute lim x→−1^− ⌊x⌋, lim x→−1^+ ⌊x⌋,lim x→2^− ⌊x⌋, and lim x→2^+ ⌊x⌋.

Verified step by step guidance

1. Understand the floor function \( \lfloor x \rfloor \), which returns the greatest integer less than or equal to \( x \).
2. For \( \lim_{x \to -1^-} \lfloor x \rfloor \), consider values of \( x \) approaching \(-1\) from the left. The floor function will return \(-2\) for values like \(-1.1\), \(-1.5\), etc.
3. For \( \lim_{x \to -1^+} \lfloor x \rfloor \), consider values of \( x \) approaching \(-1\) from the right. The floor function will return \(-1\) for values like \(-0.9\), \(-0.5\), etc.
4. For \( \lim_{x \to 2^-} \lfloor x \rfloor \), consider values of \( x \) approaching \(2\) from the left. The floor function will return \(1\) for values like \(1.9\), \(1.5\), etc.
5. For \( \lim_{x \to 2^+} \lfloor x \rfloor \), consider values of \( x \) approaching \(2\) from the right. The floor function will return \(2\) for values like \(2.1\), \(2.5\), etc.

Key Concepts

Floor Function

The floor function, denoted as ⌊x⌋, returns the largest integer that is less than or equal to a given real number x. For example, ⌊3.7⌋ equals 3, while ⌊-2.3⌋ equals -3. Understanding this function is crucial for evaluating limits involving discontinuities at integer values.

Limits from the Left and Right

In calculus, the limit of a function as x approaches a certain value can be evaluated from the left (denoted as lim x→c^−) or from the right (denoted as lim x→c^+). These one-sided limits help determine the behavior of functions at points of discontinuity, which is essential for analyzing the floor function at integer boundaries.

Discontinuity

A function is said to be discontinuous at a point if there is a sudden jump in its value. The floor function exhibits jump discontinuities at integer values, where the output changes abruptly. Recognizing these discontinuities is vital for correctly computing limits at points where the floor function is evaluated.