91Ó°ÊÓ

In calculus, the concept of the limit of a function is central to understanding continuity. When we talk about the limit of a function, particularly as it approaches a specific point, we're exploring the behavior of the function as the input gets closer and closer to that particular point. This is like observing the path an object takes as it approaches a target. The formal way to express this is by saying: \( \lim_{{x \to a}} f(x) = L \). This notation means that as \( x \) gets arbitrarily close to \( a \), \( f(x) \) approaches the value \( L \). The limit must exist and be the same when approached from any direction for continuity at a point. In simple cases, like linear functions, the limit can often be found by direct substitution.

For a function to be continuous at a particular point, the first step is to ensure that the function is defined at that point. A function is defined at a point \( a \) if you can plug \( a \) into the function and get a real number as an output.In our original exercise, we checked if \( g(x) = x + 3 \) is defined at \( x = 2 \). By calculating \( g(2) = 2 + 3 = 5 \), we confirmed that the function is indeed defined at this point. Whenever you're assessing continuity, ensure that the function provides a valid output where you're checking. This is crucial because an undefined point would mean discontinuity right off the bat.

Linear functions are among the simplest types of functions you'll encounter in calculus. They are linear equations of the form \( f(x) = mx + b \), where \( m \) and \( b \) are constants. These functions graph as straight lines and have unique properties, such as having no breaks, jumps, or holes. Because of this, linear functions are continuous everywhere on their domain. In the exercise, since \( g(x) = x + 3 \) is a linear function, it intuitively follows that it is continuous for all \( x \). Hence, there is no need for complicated calculations to verify its continuity at \( x=2 \) or elsewhere.