Use the graph of the greatest integer function y=⌊x⌋, Figure 1.10 in Section 1.1, to help you fi

step 1 Okay, so we have the limits as data approaches three from the right of the absolute valid data,

Use the graph of the greatest integer function y=⌊x⌋, Figure...

Use the graph of the greatest integer function $y=\lfloor x\rfloor,$ Figure 1.10 in Section $1.1,$ to help you find the limits. \begin{equation}\quad \text { a. }\lim _{\theta \rightarrow 3^{+}} \frac{\lfloor\theta\rfloor}{\theta} \quad \text { b. } \lim _{\theta \rightarrow 3 ^{-}} \frac{\lfloor\theta\rfloor}{\theta}\end{equation}

Use the graph of the greatest integer function $y=\lfloor x\rfloor,$ Figure 1.10 in Section $1.1,$ to help you find the limits.
\begin{equation}\quad \text { a. }\lim _{\theta \rightarrow 3^{+}} \frac{\lfloor\theta\rfloor}{\theta} \quad \text { b. } \lim _{\theta \rightarrow 3 ^{-}} \frac{\lfloor\theta\rfloor}{\theta}\end{equation}

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Key Concepts

One-Sided Limits

One-sided limits deal with the value a function approaches as the independent variable approaches a specific point from one side only (either the left or the right). This concept is crucial in the context of functions that have different behaviors on either side of a point, helping to determine the existence and value of limits in cases involving discontinuities or piecewise definitions.

Absolute Value Function

The absolute value function, denoted by |x|, gives the non-negative value of x regardless of its sign. Its piecewise definition, where the output equals x for non-negative inputs and -x for negative inputs, plays a key role in limit calculations, particularly when considering the behavior of functions as they approach a specific point.

Greatest Integer (Floor) Function

This function, commonly denoted by ?x?, returns the greatest integer less than or equal to any given real number. Its stepwise nature, with jumps (discontinuities) at integer values, makes it important to understand when evaluating one?sided limits, especially near points where the function value changes abruptly.

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