StellarTables help us estimate a limit by showing how the output values of a function behave as the input gets closer and closer to a target number. Instead of looking at a graph, we look at numerical patterns. The key idea is the same: we are not mainly interested in the value of the function exactly at the point, but rather in what the function values are approaching from nearby x-values. To do this well, we should always check values from both the left side and the right side of the target x-value.
When reading a table, look for whether the outputs appear to settle toward one number, two different numbers, or no clear value at all. If the left-side and right-side values both approach the same number, then the two-sided limit exists and equals that number. If the two sides approach different numbers, then the two-sided limit does not exist. A table can also suggest that the function is increasing or decreasing without bound, which may indicate an infinite limit. Tables are especially useful when a function is complicated, when no graph is given, or when we want numerical evidence before doing algebra.
Example 1: Use the table to estimate .
Example 1: Use the table to estimate
As x approaches 2 from the left, the values of f(x) get closer to 5. As x approaches 2 from the right, the values of f(x) also get closer to 5. Since both sides approach the same number, we estimate
Example 2:
Use the table to estimate .
As x approaches 3 from the left, the values of f(x) approach 1. As x approaches 3 to the right, the values of f(x) approach 5. Since the left-hand limit and right-hand limit are different, the two-sided limit does not exist. Therefore,