HOW TO FIND A LIMIT USING A TABLE

About "How to Find a Limit Using a Table"

How to Find a Limit Using a Table :

Here we are going to see how to find a limit using a table.

Before look into example problems, first let us see the meaning of the word "Limit"

Let I be an open interval containing x∈ R. Let f : I -> R. Then we say that the limit of f(x) is L, as x approaches x0 [Usually written as lim x -> 0 f(x)  =  L], if, whenever x becomes sufficiently close to x0 from either side with x ≠ xgets sufficiently close to L. Question 1 :

Complete the table using calculator and use the result to estimate the limit.

lim x->2 (x - 2)/(x2 - x - 2) Solution :

Instead of applying the values of x directly in the given function, we may simplify the function and apply the values of x one by one given in the table.

lim x->2 (x - 2)/(x2 - x - 2)

Since we have a quadratic function in the denominator, we may find factors and simplify.

=  lim x->2 (x - 2)/(x - 2)(x + 1)

f(x)  =  lim x->2 1/(x + 1)

 xif x = 1.9 f(x) = lim x->2 1/(x + 1)f(1.9)  =  1/(1+1.9)=  1/2.9=  0.3448 if x = 1.99 f(1.99)  =  1/(1+1.99)=  1/2.99=  0.33444 if x = 1.999 f(1.999)  =  1/(1+1.999)=  1/2.999=  0.33344 if x = 2.001 f(2.001)  =  1/(1+2.001)=  1/3.001=  0.33322 if x = 2.01 f(2.001)  =  1/(1+2.01)=  1/3.01=  0.33222 if x = 2.1 f(2.1)  =  1/(1+2.1)=  1/3.1=  0.3225

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.

Question 2 :

lim x->2 (x - 2)/(x2 - 4) Solution :

lim x->2 (x - 2)/(x2 - 4)  =  lim x->2 (x - 2)/(x + 2)(x - 2)

=  lim x->2 1/(x + 2)

 xif x = 1.9 f(x) = lim x->2 1/(x + 2)f(1.9)  =  1/(1.9+2)=  1/3.9=  0.2564 if x = 1.99 f(1.99)  =  1/(1.99 + 2)=  1/3.99=  0.2506 if x = 1.999 f(1.999)  =  1/(1.999 + 2)=  1/3.999=  0.2500 if x = 2.001 f(2.001)  =  1/(2.001+2)=  1/4.001=  0.2499 if x = 2.01 f(2.001)  =  1/(2.01+2)=  1/4.01=  0.2493 if x = 2.1 f(2.1)  =  1/(2.1+2)=  1/0.2439=  0.3225

Here x->2 appears between 1.999 to 2.001. By observing the table, we may estimate the limit as 0.25

Question 3 :

lim x -> 0 (√(x + 3) - √3)/x Solution :

=  lim x -> 0 (√(x + 3) - √3)/x

 xif x = -0.1 f(x) =  lim x -> 0 (√(x + 3) - √3)/x=  (√(-0.1+3)-√3)/(-0.1)=  (√(2.9) - √3)/(-0.1)=  0.2911 if x = -0.01 f(-0.01)  =  (√(-0.01+3)-√3)/(-0.01)=  (√(2.9) - √3)/(-0.01)=  0.2889 if x = -0.001 f(-0.001) = (√(-0.001+3)-√3)/(-0.001)=  (√(2.999) - √3)/(-0.001)=  0.2887 if x = 0.001 f(0.001) = (√(0.001+3)-√3)/(0.001)=  (√(3.001) - √3)/(0.001)=  0.28865 if x = 0.01 f(0.01) = (√(0.01+3)-√3)/(0.01)=  (√(3.01) - √3)/(0.01)=  0.2866 if x = 0.1 f(0.1) = (√(0.1+3)-√3)/(0.1)=  (√(3.1) - √3)/(0.1)=  0.2863

Here x->0 appears between -0.001 to 0.001. By observing the table, we may estimate the limit as 0.288 After having gone through the stuff given above, we hope that the students would have understood, "How to Find a Limit Using a Table"

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