Problem 46 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find each of the following limits or state that it does not exist. (a) \(\lim _{x \rightarrow 3}[x] / x\) (b) \(\lim _{x \rightarrow 0^{+}} \llbracket x \rrbracket / x\) (c) \(\lim _{x \rightarrow 1.8}[x]\) (d) \(\lim _{x \rightarrow 1.8}[x] / x\)

Step-by-Step Solution

Answer

(a) \(\frac{2}{3}\); (b) 0; (c) 1; (d) \(\frac{1}{1.8}\).

Step 1: Analyze Part (a)

The function given is \([x] / x\) as \(x\) approaches 3. \([x]\) is the greatest integer function, which outputs the largest integer less than or equal to \(x\). Therefore, as \(x\) approaches 3 from both sides (left and right), \([x]\) will be 2. As a result, the expression \([x] / x\) simplifies to \(2/3\) when \(x\) is near 3. Thus: \[\lim _{x \rightarrow 3}\frac{[x]}{x} = \frac{2}{3}\]

Step 2: Analyze Part (b)

The expression here is \(\llbracket x \rrbracket / x\) as \(x\) approaches 0 from the right. \(\llbracket x \rrbracket\) represents the floor function, which is equivalent to \([x]\). As \(x\) approaches 0 from the positive side, \([x]\) is 0. Hence, the function simplifies to \(0/x\), which is 0 for all \(x eq 0\). Thus:\[\lim _{x \rightarrow 0^{+}} \frac{\llbracket x \rrbracket}{x} = 0\]

Step 3: Analyze Part (c)

The function given is \([x]\) as \(x\) approaches 1.8. The greatest integer not greater than 1.8 is 1. Hence:\[\lim _{x \rightarrow 1.8}[x] = 1\]

Step 4: Analyze Part (d)

The expression here is \([x] / x\) as \(x\) approaches 1.8. For any \(x\) near 1.8, \([x]\) evaluates to 1 because \(1 \leq x < 2\). Therefore, the expression simplifies to \(1/x\). As \(x\) approaches 1.8, the expression becomes \(1/1.8\). Hence:\[\lim _{x \rightarrow 1.8}\frac{[x]}{x} = \frac{1}{1.8}\]

Key Concepts

Greatest Integer Function

The greatest integer function, denoted as \([x]\), is a type of step function. It's also frequently referred to as the floor function in mathematics. Its job is to return the largest integer that is less than or equal to a given number \(x\). Imagine you have any real number, this function simply rounds down to the nearest whole number. For example, if \(x = 3.9\), then \([x] = 3\), because 3 is the largest whole number that is less than or equal to 3.9.
Some key points:

• For any integer \(n\), \([n] = n\).
• For non-integers, the function "floors" the number to the next lowest integer.

This function is piecewise constant, and you observe jumps as you move through values of \(x\) across integers. This makes it particularly interesting when evaluating limits in calculus.

Floor Function

The floor function, symbolized as \(\llbracket x \rrbracket\), performs the same operation as the greatest integer function; it returns the greatest integer less than or equal to a given number. This notation is widely accepted in mathematics. Hence, both \([x]\) and \(\llbracket x \rrbracket\) essentially describe the same operation: rounding down to the nearest integer.
For instance:

• \(\llbracket 2.3 \rrbracket = 2\)
• \(\llbracket -1.7 \rrbracket = -2\)

The floor function is essential when dealing with piecewise-defined mathematical problems, as it allows us to break continuous domains into discrete intervals. This characteristic becomes particularly valuable when analyzing limits, where each small interval can often yield a different output.

Limit Calculations

Limit calculations are a cornerstone of calculus, allowing us to understand the behavior of a function as its input approaches a specific point. In the context of the greatest integer function or floor function, calculating limits involves examining the output as \(x\) gets very close to a critical point.
Due to the step-like nature of these functions, the limit often involves determining what the integer output is as \(x\) approaches from either side. For example, when evaluating: - \(\lim _{x \rightarrow 3}\frac{[x]}{x}\), as detailed in the exercise above, we check what integer \([x]\) values become as \(x\) nears 3. Since \([x]\) is constant over intervals, these calculations provide insight into the behavior right up to, but not always including, the integer.
Limit calculations help emphasize the importance of approaching the point from both the lower and the upper direction, highlighting potential discontinuities.

Approaching a Point

Approaching a point in calculus is akin to zooming in on values near a specific number \(x\) and analyzing how a function behaves. This concept is vital when working with limits because it teaches us about continuity, gaps, or jumps in function behavior.
Here are some illustrative points:

• Approaching from the right, such as \(x \rightarrow 0^{+}\), signifies considering values that are slightly greater than the point 0.
• On the other hand, approaching from the left, like \(x \rightarrow 3^{-}\), indicates we are thinking about numbers just below 3.

This directional approach is crucial when evaluating limits involving floor or greatest integer functions. Since these functions create jump discontinuities at integer values, it's essential to define clearly how \(x\) is approaching (from the right or the left), ensuring our limit evaluations accurately reflect the function's behavior at those points."