91Ó°ÊÓ

Linear functions are fundamental in mathematics and play a significant role in calculus. A linear function can be expressed in the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. This function results in a straight line graph and is known for its simplicity and predictability.

• The slope \( m \) determines the steepness or direction of the line. If \( m > 0 \), the line slopes upwards, and if \( m < 0 \), it slopes downwards.
• The y-intercept \( b \) is the point where the line crosses the y-axis. It indicates the value of \( f(x) \) when \( x = 0 \).
• The defining characteristic of linear functions is that they have constant rates of change. This leads to straight-line graphs.

In our exercise, we deal with the linear function \( f(x) = 3x + 2 \). Here, the slope is 3, and the y-intercept is 2, meaning the line crosses the y-axis at point (0, 2) and rises three units vertically for every one unit it moves horizontally.

Continuity is an important concept in calculus, especially when dealing with limits. A function is said to be continuous at a point if there are no breaks, jumps, or holes at that point. For linear functions such as \( f(x) = 3x + 2 \), continuity is inherent because:

• They are defined for all real numbers and have no gaps in their domain.
• Their graphs are unbroken straight lines.
• For a function to be continuous at a specific value \( a \): the function must be defined at \( a \), the limit of the function as \( x \) approaches \( a \) must exist, and these two values must be the same.

In our example, \( f(x) = 3x + 2 \) is continuous everywhere, particularly at \( x = 1 \), because it is a linear function. As such, the function behaves predictably, showing no abrupt changes at \( x = 1 \). This predictability aids in evaluating the limits smoothly.

Evaluating limits involves finding out what value a function approaches as the input approaches a specific point. In calculus, limits are crucial for understanding the behavior of functions at specific points, especially those related to continuity and differentiability. To evaluate the limit of a function as \( x \) approaches a particular value:

• Substitute values that converge towards the point of interest from both sides, known as left-hand and right-hand limits.
• Calculate the value of the function at these points and observe the trend as \( x \) gets closer.
• If both the left-hand and right-hand tendencies point towards a single value, the limit exists and equals that value.

In our exercise with \( f(x) = 3x + 2 \), as \( x \) approaches 1 either from the left (e.g., \( x = 0.999 \)) or from the right (e.g., \( x = 1.001 \)), the corresponding \( f(x) \) values approach 5. Therefore, the limit is \( 5 \) as \( x \) approaches 1, demonstrating the smoothness and predictability of linear functions in evaluating limits.