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Limits are a fundamental concept in calculus. They help us understand how a function behaves as the input approaches a certain value. When we talk about a limit, we're interested in what the function is approaching, even if it never actually reaches that value.
For instance, if we have a function where the input approaches -9, we want to know what value the function itself is approaching as we get closer and closer to -9.

• Limits can exist even if the function is not defined at a specific point.
• They essentially smooth out the behavior of functions as inputs get very close to a given number.
• We use the notation \( \lim_{x \rightarrow c} f(x) \) to denote the limit of a function \( f(x) \) as \( x \) approaches \( c \).

Understanding limits give us a way to predict and analyze the behavior of functions, especially when direct evaluation isn't possible.

Linear functions are among the simplest types of functions. They represent a constant rate of change, which means they graph as straight lines.
The general form of a linear function is \( f(x) = mx + b \), where:

• \(m\) is the slope, telling us how steep the line is.
• \(b\) is the y-intercept, indicating where the line crosses the y-axis.

For the function \(5x\), the slope is 5, and the y-intercept is 0, making the graph a straight line through the origin.
Linear functions are continuous, meaning there are no breaks or holes in their graphs. This continuity makes calculating limits straightforward because we can often substitute the value directly into the expression, as we would in substituting -9 into \(5x\).
Linear functions are foundational in calculus because they model many real-world situations with a constant rate of change.

When evaluating the limit of a linear function, the process is usually direct and simple. Since linear functions are continuous, we can find limits by straightforward substitution.

• Step 1: Identify the function - In this case, \(5x\).
• Step 2: Substitute the given value approaching -9 into the function.
• Step 3: Calculate \(5(-9)\), which results in -45.

The limit of the function as \(x\) approaches -9 is -45. Since the function \(5x\) is linear and continuous, the limit at any point \(c\) is simply the value of the function at that point. Realizing this can make evaluating limits straightforward and manageable, eliminating any need for complex algebra or calculus tricks. Linear functions offer the perfect introduction to the broader topic of limits in calculus.