Here we are going to see how to find a limit using a table.
First let us understand what is limit.
Let I be an open interval containing x_{0 }∈ R.
Let f : I ---> R. Then we say that the limit of f(x) is L, as x approaches x_{0} [Usually written as lim_{x--->0}f(x) = L]. Whenever x becomes sufficiently close to x_{0} from either side with x ≠ x_{0}, f(x) gets sufficiently close to L.
In the following examples, complete the table using calculator and use the result to estimate the limit.
Example 1 :
Solution :
Substitute the values of x from the table above into f(x).
When x = 1.9,
When x = 1.99,
When x = 1.999,
When x = 2.001,
When x = 2.01,
When x = 2.1,
From the above table, we have to estimate the limit when x tends to 2.
Here x--->2 appears between 1.999 to 2.001. By observing the table, we may estimate the limit as 0.333.....
Example 2 :
Solution :
Substitute the values of x from the table above into f(x).
When x = 1.9,
When x = 1.99,
When x = 1.999,
When x = 2.001,
When x = 2.01,
When x = 2.1,
Here x--->2 appears between 1.999 to 2.001. By observing the table, we may estimate the limit as 0.25.
Example 3 :
Solution :
When x = -0.1,
When x = -0.01,
When x = -0.001,
When x = 0.001,
When x = 0.01,
When x = 0.1,
Here x--->0 appears between -0.001 to 0.001. By observing the table, we may estimate the limit as 0.288.....
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