Problem 34 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(31 - 36 ,\) determine the limit. $$\lim _ { x \rightarrow 2 ^ { - } } \operatorname { int } x$$

Step-by-Step Solution

Answer

The limit of the function as x approaches 2 from the left is 1.

Step 1: Understand the Function

The function given is the integer part function (floor function). This function gives the largest integer less than or equal to x. So, for any integer value x, \(\operatorname { int } x = x\). And for any non-integer x, \(\operatorname { int } x\) is the integer part of x, which is the largest integer less than x.

Step 2: Apply the Limit

The limit is given as \(x \rightarrow 2^-\), indicating that x is approaching 2 from the left or the negative side. So, the values of x are slightly less than 2, but not equal to 2. When these values are placed into the integer part function, we get integers less than 2.

Step 3: Determine the Limit

As x is approaching 2 from the left side, all the numbers being considered are less than 2 but very close to 2. So the greatest integer value less than these fractions would be 1. Hence, the limit of the function as x approaches 2 from the left side is 1.

Key Concepts

Floor Function

The floor function, commonly denoted as \( \lfloor x \rfloor \), is a fascinating mathematical concept that can often intrigue learners. It is designed to return the largest integer that is less than or equal to a given number \( x \). This means:

• If \( x = 3.7 \), then \( \lfloor x \rfloor = 3 \).
• If \( x = -1.2 \), then \( \lfloor x \rfloor = -2 \).
• If \( x = 5 \), which is already an integer, \( \lfloor x \rfloor = 5 \).

This function plays a crucial role in various mathematical operations and is particularly important when dealing with discrete structures. Whenever you see \( \lfloor x \rfloor \), remember it simply "rounds down" to the nearest lower integer. This rounding behavior makes it the go-to function when you need to handle real numbers and integers simultaneously.

One-Sided Limit

A one-sided limit is a type of limit that considers only values of a function as the independent variable approaches a specific point from one side — either the right or the left. In mathematical terms, the notation \( \lim_{x \to a^-} f(x) \) refers to the limit of \( f(x) \) as \( x \) approaches \( a \) from the left side.

• The left-hand limit looks at values less than \( a \) but getting closer to \( a \).
• Conversely, the right-hand limit would consider values greater than \( a \).
• It's common in cases where the function behaves differently when approached from different sides of a point.

One-sided limits are crucial when exploring points where a function may not be continuous, or when only a specific side of behavior is relevant to your calculation. This approach ensures precision, especially in scenarios where you encounter a jump, discontinuity, or any point-specific behavior.

Approaching from the Left

When a function's behavior is examined as its variable approaches a certain value "from the left," we're interested in its left-hand limit. As per the notation, \( x \to 2^- \) implies we're looking at values where \( x \) is slightly less than 2, getting as close as possible without reaching or exceeding it.

• In practice, values like 1.9, 1.99, or 1.999 can be examples of \( x \) approaching 2 from the left.
• It's vital in unearthing the behavior of \( f(x) \) just before it might change upon reaching a particular point.
• Such analysis is useful when dealing with floor functions, as demonstrated by \( \lfloor x \rfloor \) which rounds down numbers just shy of an integer to the preceding integer.

Understanding the concept of approaching from the left helps appreciate subtle changes in function behavior, offering deep insights into both continuous and discrete mathematical realms.