91Ó°ÊÓ

Understanding continuity is essential when discussing limits. A function is continuous at a point if there is no interruption in its graph at that point. More precisely, a function \( f(x) \) is continuous at \( x = c \) if the following three conditions are met:

• \( f(c) \) is defined.
• \( \lim_{x \to c} f(x) \) exists.
• \( \lim_{x \to c} f(x) = f(c) \).

If any of these conditions fail, the function is not continuous at \( c \). For the problem at hand, the function \( f(x) \) may not be continuous at \( x = 3 \) since \( f(x) \) is defined as \( g(x) \) for \( x eq 3 \), and its value at \( x = 3 \) is not explicitly given. Despite this, the limit still exists because the value directly at the point doesn’t affect the limit, as long as \( g(x) \) approaches a specific value. In this case, \( f(x) \) approaches \( 4 \) as \( x \) approaches \( 3 \), so the limit exists at that point even if continuity is not guaranteed.

Piecewise functions are functions defined by different expressions over different parts of their domain. The function in our exercise is given by \[f(x) = \begin{cases} g(x), & \text{if } x eq 3 \ ? , & \text{if } x = 3\end{cases}\]This kind of function is especially useful in mathematical problems where different rules apply at different intervals. In this example, we focus on \( x eq 3 \) where \( f(x) = g(x) \).
This allows us to evaluate the limit without needing the specific value of \( f(3) \). Piecewise functions are all about observing behavior just before and after a point, which is why limits are concerned with the values that the function approaches from either side of a certain point.

Substitution is a common method in limits, especially when dealing with functions like those in this exercise. The principle involved is straightforward: if two functions are equal everywhere except possibly at a single point, their limits are also equal as they approach that point.
In the exercise, since \( f(x) = g(x) \) for every \( x eq 3 \), and given that \( \lim_{x \to 3} g(x) = 4 \), substitution allows us to conclude that:\[\lim_{x \to 3} f(x) = \lim_{x \to 3} g(x) = 4\]Using substitution simplifies the calculation of limits, as it allows you to swap a complex function for a simpler one when analyzing intuitive behavior around specific points.
It’s a powerful tool because it respects the nature of limits, focusing on approaching values rather than fixed values at particular points.