The correct notation for a limit is given as limx→af(x) = L, where x is the independent variable, a is the value the independent variable approaches, f(x) is the function, and L is the limit value.
The arrow in the limit notation represents the direction in which the independent variable x approaches the value a. In this case, x approaches a from both sides, meaning it can approach a from values greater than or less than a.
A one-sided limit, denoted as limx→a⁺f(x) or limx→a⁻f(x), only considers the values of x approaching a from one side (either the right or the left). A two-sided limit, denoted as limx→af(x), considers values of x approaching a from both sides.
If a limit does not exist, it means that the function f(x) does not approach a specific value as x approaches a. This could happen if the function has a vertical asymptote or if the values of f(x) approach different values from each side.
No, a limit cannot be evaluated by simply plugging in the value of a for x. This is because a limit considers the behavior of the function f(x) as x approaches a, not necessarily at the value x = a. It is possible for the limit to exist even if the function is undefined at x = a.