Computing limits by the method of making a table of values

This method is generally only used to express the notion of a limit, and not to compute limits, as it is quite a lengthy one:

For example, there is a function `f(x) = 2x^2 + 3x + 1`. In order to find its left and right limits as x approaches 1, you need to proceed as follows:

Left limit as x approaches 1:

A limit is talking about the general value which the function is approaching as x approaches a certain value. In hes 1. Its left limit is the general value which the function takes to the left of x = 1. Thus, in order to see where the function is going without a graph, we assume an x value to the left of 1 (smaller than 1), say x = 0.1, and move closer to 1 in small steps of, say, 0.01. Evaluating the function at each of these x values, we get the following table:

x value	Function value
0.90	f(0.90) = 5.32
0.91	f(0.91) = 5.3862
0.92	f(0.92) = 5.4528
0.93	f(0.93) = 5.5198
0.94	f(0.94) = 5.5872
0.95	f(0.95) = 5.655
0.96	f(0.96) = 5.7232
0.97	f(0.97) = 5.7918
0.98	f(0.98) = 5.8608
0.99	f(0.99) = 5.9302
1.00	f(1.00) = 6.0

Left limit as x approaches 1

It is clear that the function values are approaching 6 as x is approaching 1. Thus the left limit of `f(x) = 2x^2 + 3x + 1` as x approaches 1 is 6. This is written as `lim_(x -> 1^-) f(x) = 2x^2 + 3x + 1`

Similarly, the right limit is obtained by taking an x value just to the right of x = 1 and then moving closer and closer towards x = 1 from it. Evaluating the function at each x value, we get the following table for the right limit:

Right limit as x approaches 1

x value	Function value
1.10	f(1.1) = 6.72
1.01	f(1.09) = 6.6462
1.02	f(1.08) = 6.5728
1.03	f(1.07) = 6.4998
1.04	f(1.06) = 6.4272
1.05	f(1.05) = 6.355
1.06	f(1.04) = 6.2832
1.07	f(1.03) = 6.2118
1.08	f(1.02) = 6.1408
1.09	f(1.01) = 6.0702
1.00	f(1.00) = 6.0

Right limit as x approaches 1

The function values are approaching 6 as x is approaching 1 from the right hand side. So the right hand limit of the function as x approaches x is 6, `lim_(x -> 1^+) f(x) = 6`

Normal limit as x approaches 1

Since both the left and right limits are equal, then the normal limit of the function is equal to them. So the normal limit of the function as x is approaching 1 is 6. That is `lim_(x -> 1) f(x) = 6`

The above method of finding the limits of a function is used to demonstrate the definition of the limits of a function for any value of x

The limit of a function as x approaches a particular value is the value that the function is approaching around x.