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 x0 ∈ 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 ≠ x0 gets 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.
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