This tutorial: Part A: Numerical Approach 
Next tutorial: Part B: Graphical Approach 
Estimating Limits Numerically
Look at the function
f(x)  = 
x  2 
and ask yourself: "What happens to f(x) as x approaches 2?" (Notice that you cannot simply substitute x = 2, because the function is not defined at x = 2.) The following chart shows the value of f(x) for values of x close to, and on either side of 2:
x approaching 2 from the left  x approaching 2 from the right  


We have left the entry under 2 blank to emphasize that, when calculating the limit of f(x) as x approaches 2, we are not interested in its value when x equals 2. Notice from the table that, the closer x gets to 2 from either side, the closer f(x) gets to 12. We write this as:
x→2  f(x) = 12 
In words:
Q What if we had gotten different answers when approaching 2 from the left and right?
A Suppose, for instance, that the table looked like this:
x approaching 2 from the left  x approaching 2 from the right  


Notice that the limit appears to be 12 as you approach from the left, but it now appears to be 4^{1}/3 if you approach from the right. We therefore write:
x→2^{}  g(x) = 12  The limit of g(x), as x approaches 2 from the left, equals 12  
and  
x→2^{}  g(x) = 4^{1}/3  The limit of g(x), as x approaches 2 from the right, equals 4^{1}/3 
Before going on to the first practice question, look over the following summary of terms.
Definition of a Limit
If the left limit and the right limit exist and are equal (to L, say) then we say that lim_{x → a}f(x) exists and equals L, and write

First calculate the missing values in the following table (might we suggest you use the Function Evaluator & Grapher for this) and then decide on a numerical estimate of lim_{x → 3}f(x). 
x approaching 3 from the left  x approaching 3 from the right  


Q  x→3  f(x) = ? 
7.0  5.0  3.0  
There is no limit; the numbers on the left are big positive numbers, while those on the right are big negative numbers 
Suppose a certain function g has the following table of values: