51. The value of:
limx→2 (x3 − 8)/(x − 2)
is _______. (in integer)
Evaluate the Limit of (x³ − 8)/(x − 2) as x Tends to 2
Understanding the Given Limit Problem
This question is based on the evaluation of an algebraic limit. The given expression is:
limx→2 (x3 − 8)/(x − 2)
To evaluate a limit, the first step is usually to substitute the limiting value directly into the expression. If direct substitution gives a definite real number, that number is generally the value of the limit. However, if direct substitution produces an indeterminate form such as 0/0, the expression must first be simplified.
In this problem, direct substitution gives the indeterminate form 0/0. Therefore, we need to factorize the numerator using the algebraic identity for the difference of two cubes.
Checking the Limit by Direct Substitution
The given limit is:
limx→2 (x3 − 8)/(x − 2)
Substituting x = 2 into the numerator:
23 − 8 = 8 − 8 = 0
Substituting x = 2 into the denominator:
2 − 2 = 0
Therefore, direct substitution gives:
0/0
The expression 0/0 is an indeterminate form. It does not mean that the answer is zero, nor does it mean that the limit does not exist. It indicates that the original expression must be simplified before the limiting value can be determined.
Using the Difference of Cubes Identity
The numerator is:
x3 − 8
Since:
8 = 23
we can rewrite the numerator as:
x3 − 23
Now, use the standard algebraic identity:
a3 − b3 = (a − b)(a2 + ab + b2)
Taking:
a = x
and:
b = 2
we obtain:
x3 − 23 = (x − 2)(x2 + 2x + 4)
Therefore:
x3 − 8 = (x − 2)(x2 + 2x + 4)
Simplifying the Given Limit
Substituting the factorized form of the numerator into the original expression gives:
limx→2 [(x − 2)(x2 + 2x + 4)]/(x − 2)
The factor (x − 2) appears in both the numerator and denominator. For values of x ≠ 2, this common factor can be cancelled.
Therefore, the limit becomes:
limx→2 (x2 + 2x + 4)
The indeterminate form has now been removed, and the remaining expression is a polynomial. Since polynomial functions are continuous, we can directly substitute x = 2.
Substituting x = 2 Into the Simplified Expression
We now evaluate:
limx→2 (x2 + 2x + 4)
Substituting x = 2:
= 22 + 2(2) + 4
Calculating each term:
22 = 4
2(2) = 4
Therefore:
= 4 + 4 + 4
Hence:
= 12
Therefore, the value of the given limit is 12.
Why Cancelling the Factor (x − 2) Is Valid
It may appear that cancelling (x − 2) is not allowed because the limit is taken as x → 2. However, a limit describes the behaviour of a function when x approaches a particular value; it does not require x to be exactly equal to that value during the simplification.
For all values of x close to 2 but not equal to 2:
[(x − 2)(x2 + 2x + 4)]/(x − 2) = x2 + 2x + 4
Therefore, both expressions have exactly the same values in a neighbourhood around x = 2, except possibly at x = 2 itself. Since a limit depends on nearby values, the cancellation is valid for evaluating the limit.
Alternative Solution Using the Standard Limit Formula
The given expression can also be evaluated using the standard result:
limx→a (xn − an)/(x − a) = nan−1
In the given problem:
x3 − 8 = x3 − 23
Therefore:
n = 3
and:
a = 2
Applying the formula:
limx→2 (x3 − 23)/(x − 2) = 3(2)3−1
Therefore:
= 3(22)
= 3 × 4
Hence:
= 12
This confirms the result obtained by the factorization method.
Alternative Interpretation Using the Derivative
The limit can also be recognized as the definition of the derivative. The derivative of a function f(x) at x = a is defined as:
f′(a) = limx→a [f(x) − f(a)]/(x − a)
Let:
f(x) = x3
Then:
f(2) = 23 = 8
Therefore, the given limit becomes:
limx→2 [f(x) − f(2)]/(x − 2)
This is exactly:
f′(2)
Since:
f′(x) = 3x2
we obtain:
f′(2) = 3(22)
= 3 × 4
Therefore:
f′(2) = 12
Thus, the derivative method also confirms the answer.
Final Answer
The given limit is:
limx→2 (x3 − 8)/(x − 2)
Using the difference of cubes:
x3 − 8 = (x − 2)(x2 + 2x + 4)
Therefore:
limx→2 (x2 + 2x + 4)
= 4 + 4 + 4
Hence:
12
Correct Answer: 12
