Step 1: We'd like to pick a value that's a little bit less than x = 2 x=2x, equals, 2 (that is, a value that's "to the left" of 2 22 when thinking about the standard x xx-axis), so maybe start with something like x = 1.9 x=1.9x, equals, 1, point, 9. x xx 1.9 1.91, point, 9 2 22 f ( x ) f(x)f, left parenthesis, x, right parenthesis 0.2564 0.25640, point, 2564 undefined Step 2: Try a couple more x xx-values to simulate the feeling of getting infinitely close to x = 2 x=2x, equals, 2 from the left. x xx 1.9 1.91, point, 9 1.99 1.991, point, 99 1.9999 1.99991, point, 9999 2 22 f ( x ) f(x)f, left parenthesis, x, right parenthesis 0.2564 0.25640, point, 2564 0.2506 0.25060, point, 2506 0.25001 0.250010, point, 25001 undefined Notice how our x xx-values { 1.9 , 1.99 , 1.9999 } {1.9,1.99,1.9999}left brace, 1, point, 9, comma, 1, point, 99, comma, 1, point, 9999, right brace really "zoom in" around x = 2 x=2x, equals, 2. A worse choice of x xx-values would have been constant increments like { − 1 , 0 , 1 } {−1,0,1}left brace, minus, 1, comma, 0, comma, 1, right brace, which aren't very helpful for thinking about getting infinitely close to x = 2 x=2x, equals, 2. Step 3: Approach x = 2 x=2x, equals, 2 from the right just like we did from the left. We want to do this in a way that simulates the feeling of getting infinitely close to x = 2 x=2x, equals, 2. x xx 1.9 1.91, point, 9 1.99 1.991, point, 99 1.9999 1.99991, point, 9999 2.0001 2.00012, point, 0001 2.01 2.012, point, 01 2.1 2.12, point, 1 f ( x ) f(x)f, left parenthesis, x, right parenthesis 0.2564 0.25640, point, 2564 0.2506 0.25060, point, 2506 0.25001 0.250010, point, 25001 0.24999 0.249990, point, 24999 0.2494 0.24940, point, 2494 0.2439 0.24390, point, 2439 (Note: We've removed x = 2 x=2x, equals, 2 from the table to save space, and also because it isn't necessary for reasoning about the limit value.) Looking at the table we've created, we have very strong evidence that the limit is 0.25 0.250, point, 25. But, if we're honest with ourselves, we must admit that what we have is only a reasonable approximation. We can't say for sure that this is the actual value of the limit.
