ESTIMATING LIMITS FROM TABLES

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->-3 (√(1-x) - 2)/(x + 3)

Solution :

x

if x = -3.1

f(x) = lim x->-3 (√(1-x) - 2)/(x + 3)

f(-3.1)  =   (√(1+3.1) - 2)/(-3.1 + 3)

=  (√4.1 - 2)/(-0.1)

-0.24846

if x = -3.01

f(-3.01)  =   (√(1+3.01) - 2)/(-3.01 + 3)

=  (√4.01 - 2)/(-0.01)

 -0.2498

if x = -3.0

f(3.0)  =   (√(1+3) - 2)/(-3 + 3)

=  (√4 - 2)/0

 Indeterminant form

if x = -2.999

f(-2.999)=(√(1+2.999)-2)/(-2.999+3)

 -0.25001

if x = -2.99

f(-2.99)=(√(1+2.99)-2)/(-2.99+3)

 - 0.2501

if x = -2.9

f(-2.9)=(√(1+2.9)-2)/(-2.9+3)

 -0.2515

From the above table, we have to estimate the limit when x tends to -3. 

When x approaches - 3, f(x) tends to -0.25 approximately.   

Hence the answer is -0.25

Question 2 :

lim x->0 sin x/x 

Solution :

x

if x = -0.1

f(x) = lim x->0 sin x/x 

f(-0.1)  =  sin (-0.1)/(-0.1)

=  -sin (0.1)/(-0.1)

0.99833

if x = -0.01

f(x) = lim x->0 sin x/x 

f(-0.01)  =  sin (-0.01)/(-0.01)

=  -sin (0.01)/(-0.01)

 0.9998

if x = -0.001

f(x) = lim x->0 sin x/x 

f(-0.001)  =  sin (-0.001)/(-0.001)

=  -sin (0.001)/(-0.001)

 0.9999

if x = 0.001

f(x) = lim x->0 sin x/x 

f(0.001)  =  sin (0.001)/(0.001)

=  sin (0.001)/0.001

 0.9999

if x = 0.01

f(x) = lim x->0 sin x/x 

f(0.01)  =  sin (0.01)/(0.01)

 0.99998

if x = 0.1

f(x) = lim x->0 sin x/x 

f(0.1)  =  sin (0.1)/(0.1)

 0.99833

Here x->0 appears between -0.001 to 0.001. By observing the table, we may estimate the limit as 0.99

Hence the answer is 0.99

Question 3 :

lim x -> 0 (cos x - 1)/x

Solution :

x

if x = -0.1

f(x) =  lim x -> 0 (cos x - 1)/x

=  (cos(-0.1) - 1)/(-0.1)

=  (cos(0.1) - 1)/(-0.1)

0.0499

if x = -0.01

f(x) =  lim x -> 0 (cos x - 1)/x

=  (cos(-0.01) - 1)/(-0.01)

=  (cos(0.01) - 1)/(-0.01)

=  0.00499

if x = -0.001

f(x) =  lim x -> 0 (cos x - 1)/x

=  (cos(-0.001) - 1)/(-0.001)

=  (cos(0.001) - 1)/(-0.001)

=  0.000499

if x = 0.0001

f(x) =  lim x -> 0 (cos x - 1)/x

=  (cos(0.0001) - 1)/(0.0001)

=  (cos(0.0001) - 1)/(0.0001)

=  0.000049

if x = 0.01

f(x) =  lim x -> 0 (cos x - 1)/x

=  (cos(0.01) - 1)/(0.01)

=  (cos(0.01) - 1)/(0.01)

=  -0.00499

if x = 0.1

f(x) =  lim x -> 0 (cos x - 1)/x

=  (cos(0.1) - 1)/(0.1)

=  (cos(0.1) - 1)/(0.1)

=  -0.0499

Here x->0 appears between -0.001 to 0.0001. By observing the table, we may estimate the limit as 0.00049

Hence the answer is 0.

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