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Linear functions are one of the simplest kinds of functions in mathematics. They are defined by equations that form straight lines when plotted on a graph. The general form of a linear function is given by:
\[ f(x) = mx + b \]
where \( m \) is the slope of the line, and \( b \) is the y-intercept, or the point at which the line crosses the y-axis. Each change in \( x \) results in a constant change in \( f(x) \), making them both predictable and easy to work with.
Linear functions model relationships with constant rate of change, like speed or price rates. Since they can be easily predicted, they are widely used in both scientific and real-world applications. Grasping linear functions is an essential step for dealing with more complex types of functions later on.

Continuity is a crucial concept in calculus that helps us understand how functions behave as their input values change. A function is said to be continuous at a point if the function’s output does not suddenly jump or break at that point.
For a function \( f(x) \) to be continuous at a particular point \( c \), three conditions must be satisfied:

• \( f(c) \) must be defined. This means the function has a value at \( x = c \).
• The limit \( \lim_{x \to c} f(x) \) must exist.
• The limit \( \lim_{x \to c} f(x) \) must equal \( f(c) \).

Linear functions, as mentioned in the current problem, are continuous everywhere. This means as \( x \) approaches any point along the line, the output of the function also smoothly approaches the corresponding value without any gaps or interruptions. This property is very helpful, as it allows us to use simple substitution to find limits for linear functions.

The substitution method is a straightforward technique used to evaluate the limits of functions, especially when dealing with continuous ones like linear functions. When a function is continuous at a point, finding the limit as \( x \) approaches that point can be directly done by substituting the value of \( x \) into the function.
To effectively use this method, you should:

• Ensure the function is continuous at the point you are evaluating. Linear functions will always meet this condition.
• Simply replace \( x \) in the function with the value the variable approaches. For example, for the function \(-2x + 5\) as \( x \to 1\), substitute \( x = 1\).
• Simplify the function to get the limit. This helps in verifying that the limit makes sense logically and numerically.

By understanding and applying substitution effectively, you can solve many limit problems quickly and accurately. This method forms the basis for tackling more advanced calculus problems later on.