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)

x

if 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.

Hence the answer is 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)

x

if 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

Hence the answer is 0.25

Question 3 :

lim x -> 0 (√(x + 3) - √3)/x

Solution :

=  lim x -> 0 (√(x + 3) - √3)/x

x

if 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

Hence the answer is 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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