This notation represents the limit of the function g(x) as x approaches -1, and the limit value is equal to -2.
To prove this limit, we need to show that for any given value ε > 0, there exists a corresponding value δ > 0 such that whenever |x - (-1)| < δ (i.e. x is close enough to -1), then |g(x) - (-2)| < ε (i.e. g(x) is close enough to -2). This can be done using the epsilon-delta definition of a limit.
Yes, it is possible for a limit to be different than the function value at that point. This is because a limit is a measure of the behavior of a function as it approaches a specific point, not necessarily the value of the function at that point.
If the limit of g(x) as x approaches -1 is undefined, it means that the function g(x) either does not exist or is not defined at x = -1. This could be due to a discontinuity or an asymptote at that point.
To argue that the limit of g(x) as x approaches -1 is equal to -2, we can use a graph to show that as x gets closer and closer to -1, the values of g(x) also get closer and closer to -2. We can also use the graph to visually demonstrate that there are no sudden jumps or breaks in the function at x = -1, which would indicate that the limit does not exist.