# ESTIMATING LIMITS FROM TABLES

## About "Estimating Limits From Tables"

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 :

 xif 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.

Question 2 :

lim x->0 sin x/x

Solution :

 xif 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

Question 3 :

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

Solution :

 xif 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

After having gone through the stuff given above, we hope that the students would have understood, "Estimating Limits From Tables"

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