Question

For any real number \(x\), the floor function (or greatest integer function) \(\lfloor x\rfloor\) is the greatest integer less than or equal to \(x\) (see figure). a. Compute \(\lim _{x \rightarrow-1^{-}}\lfloor x\rfloor, \lim _{x \rightarrow-1^{+}}\lfloor x\rfloor, \lim _{x \rightarrow 2^{-}}\lfloor x\rfloor,\) and \(\lim _{x \rightarrow 2^{+}}\lfloor x\rfloor\) b. Compute \(\lim _{x \rightarrow 2,3^{-}}\lfloor x\rfloor, \lim _{x \rightarrow 2,3^{+}}\lfloor x\rfloor,\) and \(\lim _{x \rightarrow 2,3}\lfloor x\rfloor\) c. For a given integer \(a,\) state the values of \(\lim _{n \rightarrow \infty}\lfloor x\rfloor\) and $$ \lim _{x \rightarrow a^{+}}\lfloor x\rfloor $$ d. In general, if \(a\) is not an integer, state the values of \(\lim _{n \rightarrow \infty}\lfloor x\rfloor\) and \(\lim _{x \rightarrow a^{+}}\lfloor x\rfloor\). e. For what values of \(a\) does \(\lim _{x \rightarrow a}\lfloor x\rfloor\) exist? Explain.

Short answer

Answer: \(\lim _{x \rightarrow a}\lfloor x\rfloor\) exists only when \(a\) is not an integer. This is because, when \(a\) is an integer, the left and right-sided limits will be different integers (one less than the other), causing the limit to not exist.

Step-by-step solution

Calculating the first limit

As \(x\) approaches \(-1\) from the left \((-1^{-})\), the floor function \(\lfloor x\rfloor\) should approach the greatest integer less than or equal to \(-1\). In this case, the floor function is equal to the integer \(-2\). So, \(\lim _{x \rightarrow-1^{-}}\lfloor x\rfloor=-2\)

Calculating the second limit

As \(x\) approaches \(-1\) from the right \((-1^{+})\), the floor function \(\lfloor x\rfloor\) should approach the greatest integer less than or equal to \(-1\). In this case, the floor function is equal to the integer \(-1\). So, \(\lim _{x \rightarrow-1^{+}}\lfloor x\rfloor=-1\)

Calculating the third limit

As \(x\) approaches \(2\) from the left \((2^{-})\), the floor function \(\lfloor x\rfloor\) should approach the greatest integer less than or equal to \(2\). In this case, the floor function is equal to the integer \(1\). So, \(\lim _{x \rightarrow 2^{-}}\lfloor x\rfloor=1\)

Calculating the fourth limit

As \(x\) approaches \(2\) from the right \((2^{+})\), the floor function \(\lfloor x\rfloor\) should approach the greatest integer less than or equal to \(2\). In this case, the floor function is equal to the integer \(2\). So, \(\lim _{x \rightarrow 2^{+}}\lfloor x\rfloor=2\) b. Compute \(\lim _{x \rightarrow 2,3^{-}}\lfloor x\rfloor, \lim _{x \rightarrow 2,3^{+}}\lfloor x\rfloor,\) and \(\lim _{x \rightarrow 2,3}\lfloor x\rfloor\)

Calculating limits around 2.3

As x approaches 2.3 from the left or the right, the floor function is equal to the integer 2. Therefore, for any of the three limits, we have: \(\lim _{x \rightarrow 2,3^{-}}\lfloor x\rfloor=2\) \(\lim _{x \rightarrow 2,3^{+}}\lfloor x\rfloor=2\) \(\lim _{x \rightarrow 2,3}\lfloor x\rfloor=2\) c. For a given integer \(a,\) state the values of \(\lim _{n \rightarrow \infty}\lfloor x\rfloor\) and \(\lim _{x \rightarrow a^{+}}\lfloor x\rfloor\):

Integer a

As \(x\) approaches the integer \(a\) from the right, the floor function is equal to the integer \(a\). Therefore: \(\lim _{x \rightarrow a^{+}}\lfloor x\rfloor=a\) d. In general, if \(a\) is not an integer, state the values of \(\lim _{n \rightarrow \infty}\lfloor x\rfloor\) and \(\lim _{x \rightarrow a^{+}}\lfloor x\rfloor\):

Non-integer a

As \(x\) approaches the non-integer \(a\) from the right, the floor function is equal to the greatest integer less than or equal to \(a\). Let this integer be \(b\). Then, \(\lim _{x \rightarrow a^{+}}\lfloor x\rfloor=b\) e. For what values of \(a\) does \(\lim _{x \rightarrow a}\lfloor x\rfloor\) exist? Explain.

Existence of the limit

\(\lim _{x \rightarrow a}\lfloor x\rfloor\) exists if both \(\lim _{x \rightarrow a^{+}}\lfloor x\rfloor\) and \(\lim _{x \rightarrow a^{-}}\lfloor x\rfloor\) exist and are equal to each other. This will happen only when \(a\) is not an integer. When a is an integer, the left and right-sided limits will be different integers (one less than the other), so the limit will not exist.

Key concepts

Unidirectional Limits

When exploring the behavior of a function as it approaches a specific point from only one direction—either from the left or the right—we delve into the study of unidirectional limits. These are critical in understanding functions that are not necessarily continuous at every point. To capture this, we use notation like \(x \rightarrow c^{-}\) for approaching from the left, known as a left-hand limit, and \(x \rightarrow c^{+}\) for approaching from the right, known as a right-hand limit.

For the floor function, the unidirectional limits are particularly important at integer bounds because the function steps down or up precisely at these values. Visualizing the graph of the floor function can be quite helpful—it looks like a staircase descending from left to right, and where each step corresponds to an integer value.

Greatest Integer Function

Also known as the floor function, the greatest integer function denoted as \(\lfloor x \rfloor\) represents the largest integer less than or equal to the real number \(x\). Imagine each real number dropping down to the nearest step on the staircase—the step being the integer part.

This function creates a graphical representation that is a series of steps, where each step corresponds to an x-interval between integers. The interesting part occurs at the edges of these steps where \(x\) is an integer itself. At these exact points, the function value changes abruptly, creating the need to consider the concept of limits in order to understand the behavior around these 'jumps'.

Calculus Limits

In calculus, limits help us describe the behavior of functions as they approach specific points or even infinity. They are fundamental for defining continuity, derivatives, and integrals. The limit of a function at a particular point doesn't always have to equal the function's value at that point—if the function is even defined there. This is the case with the greatest integer function; for non-integer values of \(x\), \(\lim_{x \rightarrow c} \lfloor x \rfloor\) equals \(\lfloor c \rfloor\), but at integer values, the function value jumps, which requires examining both sides to determine if a limit exists at that point.

Understanding limits requires us to investigate both right-hand and left-hand limits. If both exist and are equal, we say the two-sided limit exists. In the context of the floor function, this concept is particularly intriguing because it reinforces the idea that limits capture behavior around points, not necessarily at those points.

Left-Hand Limit

A left-hand limit looks at the approach of \(x\) to a value from the left side. For \(\lim _{x \rightarrow c^{-}} f(x)\), we're interested in the behavior of \(f(x)\) as it gets infinitely close to, but never actually reaches, the point \(x=c\) from the left-hand side. Imagine creeping up to the edge of a cliff from the left—your behavior at the edge is our focus, not where you land after you step off.

For the floor function, this limit evaluates the greatest integer just before we reach \(x\). So if we approach an integer \(a\) from the left, the left-hand limit of the floor function will be \(a-1\). This step-like discontinuity makes left-hand limits especially critical for the floor function since it embodies the function's downward jumps.

Right-Hand Limit

Conversely, a right-hand limit looks at how a function behaves as \(x\) approaches a particular value from the right—represented as \(\lim _{x \rightarrow c^{+}} f(x)\). Visualize it as tiptoeing up to the cliff's edge from the right side. Here, we're observing what \(f(x)\) does right before \(x\) hits \(c\), surging from the right.

In the context of the floor function, as \(x\) approaches an integer \(a\) from the right, \(\lim _{x \rightarrow a^{+}} \lfloor x \rfloor = a\). The function holds steady at \(a\) just before stepping up to the next integer. This concept beautifully encapsulates the floor function's graphic behavior, where it maintains constancy until the brink where \(x\) becomes the next integer.