To decide whether lim_x→a  f(x) exists, and to find its value if it does:

Step 1. Draw the graph of f(x) either by hand or using technology, such as a graphing calculator.
Step 2. Position your pencil point (or the graphing calculator "trace" cursor) on a point of the graph to the right of x = a. In the example illustrated, we are estimating limx→-2 f(x), so we have a = -2. Therefore, we position our trace point on the graph to the right of x = -2.
Step 3. Move the point along the graph toward x = a from the right and read the y-coordinate as you go. The value the y-coordinate approaches (if any) is then the limit  lim x→a f(x). In the example we are doing, notice that the y-coordinate is approaching 2 as x approaches -2 from the right. (Press "Run" under the graph to see this happen.) Therefore,   lim x→-2 f(x) = 2.
Step 4. Repeat Steps 2 and 3, but this time starting from a point on the graph to the left of x = a, and approach x = a along the graph from the left. The value the y-coordinate approaches (if any) is then  lim x→a f(x). In the example we are doing, the y-coordinate is again approaching 2 as x approaches -2 from the left. Therefore,   lim x→-2 f(x) = 2.
Step 5. If the left and right limits both exist and have the same value L, then lim x→a f(x) exists and equals L. In our example, the left and right limits both exist and equal 2, and so  lim x→a f(x) = 2.