Use the graph of the greatest integer function y = ⌊x⌋, Figure 1.10 in Section 1.1, to help you find the limits in Exercises 21 and 22.

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b. limt→4−(t−⌊t⌋)

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Understand the greatest integer function, denoted as ⌊x⌋, which returns the largest integer less than or equal to x. For example, ⌊3.7⌋ = 3 and ⌊-2.3⌋ = -3.

The expression t - ⌊t⌋ represents the fractional part of t, which is the difference between t and the greatest integer less than or equal to t. This value is always between 0 (inclusive) and 1 (exclusive).

Consider the limit lim(t→4−)(t−⌊t⌋). The notation t→4− indicates that we are approaching 4 from the left, meaning t is slightly less than 4.

As t approaches 4 from the left, t can be expressed as 3.999... or any value slightly less than 4. In this case, ⌊t⌋ will be 3 because it is the greatest integer less than or equal to t.

Substitute ⌊t⌋ = 3 into the expression t - ⌊t⌋. As t approaches 4 from the left, the expression becomes 3.999... - 3, which simplifies to a value approaching 1. Therefore, the limit is 1.

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Greatest Integer Function

The greatest integer function, denoted as ⌊x⌋, maps a real number x to the largest integer less than or equal to x. This function creates a step-like graph where each integer value is a discontinuity. Understanding this function is crucial for analyzing how it behaves around integer points, especially when evaluating limits.

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A left-hand limit, denoted as limt→c−f(t), refers to the value that a function approaches as the input approaches a specific point c from the left side. This concept is essential for understanding how functions behave near discontinuities, such as those in the greatest integer function, and is key to solving the given limit problem.

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Piecewise functions are defined by different expressions over different intervals. The greatest integer function is a classic example, as it takes constant values over intervals between integers. Recognizing how to evaluate such functions at boundaries and within intervals is vital for calculating limits involving piecewise definitions.

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