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Limit notation provides a concise way to describe how a function behaves as its variable approaches a certain value, often infinity. In other words, it's a mathematical shorthand used to specify the end behavior of a function.

For our function, which is linear, expressed as \( m(t) = 5t - 7 \), the limit notation helps us understand what happens to \( m(t) \) as \( t \) moves towards positive or negative infinity.

• As \( t \to \infty \), the value of \( m(t) \) becomes infinitely large, and we express this behavior as \( \lim_{{t \to \infty}} m(t) = \infty \).
• Conversely, as \( t \to -\infty \), the function \( m(t) \) decreases towards negative infinity, leading to the notation \( \lim_{{t \to -\infty}} m(t) = -\infty \).

Using limit notation makes it easier to communicate these concepts without lengthy verbal explanations. It's especially useful when dealing with complex functions beyond linear ones.

Linear functions are among the simplest forms of functions, represented by the equation \( y = mx + b \). This equation can be identified by its straight-line graph.

Here are some key characteristics of linear functions:

• Slope \( m \): This number indicates the steepness and direction of the line. In our example, the slope is 5, suggesting that for every unit increase in \( t \), \( m(t) \) increases by 5.
• Y-intercept \( b \): This is where the line crosses the y-axis. For \( m(t) = 5t - 7 \), the y-intercept is -7, showing that when \( t = 0 \), \( m(t) = -7 \).

Linear functions continue indefinitely in both directions, unlike quadratic or cubic functions, which can curve back and forth. This straightforward nature makes them a go-to first step for understanding more complex functions.

Horizontal asymptotes describe a horizontal line that a graph of a function approaches but never touches as \( x \to \pm\infty \). They are typical in rational functions where the degrees of the polynomial in the numerator and denominator determine the presence of asymptotes.

For the case of our linear function \( m(t) = 5t - 7 \), there are no horizontal asymptotes. Here’s why:

• Linear functions have a constant rate of change defined by the slope \( m \). In this case, there is no leveling out at extreme values of \( x \), as the line will continually rise or fall.

In contrast, rational functions like \( y = \frac{2x^2 + 3}{x^2 + 1} \) can have horizontal asymptotes, as the terms in the numerators and denominators cancel each other out, creating a leveling effect.

Understanding whether a function has horizontal asymptotes lets us predict the long-term behavior of the function and its graph.