Calculus with Applications (10th Edition)

Reading the graph of exercise 10, the point (1,1) is marked with an empty circle, indicating that it DOES NOT BELONG to the graph, while the isolated point (1,2) is marked with full circle, indicating that it belongs to the graph. This means that the function value at x=1 is 2, f(1)=2. Observing the graph, approaching x=1 from either side, the points on the graph suggest that f(x) gets closer and closer to 1, so both one sided limits exist and equal 1, so $\displaystyle \lim_{x\rightarrow 1}f(x)$ =1. What this tells us is that the function value of x may not necessarily be equal to the limit at x, which will be discussed in the next section. Algebraically, the function f is likely to be defined piecewise, with one rule for x=1, and another for the rest of the values of x. (see example 3)