limx→−3 (2x + 6)/(x + 3)
is ______.
Evaluate the Limit of (2x + 6)/(x + 3) as x Approaches −3
Understanding the Given Limit Problem
The given question asks us to evaluate the limit of a rational algebraic expression as x approaches −3. The expression is:
limx→−3 (2x + 6)/(x + 3)
When solving a limit involving a rational expression, the first approach is usually direct substitution. If direct substitution gives a definite real number, that number is the value of the limit. However, if substitution produces an indeterminate form such as 0/0, the expression must first be simplified before the limit can be evaluated.
In this problem, direct substitution gives the indeterminate form 0/0. This does not mean that the limit is zero, and it also does not mean that the limit does not exist. Instead, it indicates that the numerator and denominator contain a common factor that should be simplified.
Step-by-Step Solution
Step 1: Apply Direct Substitution
We begin by substituting x = −3 into the given expression:
(2x + 6)/(x + 3)
Substituting x = −3 into the numerator gives:
2(−3) + 6 = −6 + 6 = 0
Substituting x = −3 into the denominator gives:
−3 + 3 = 0
Therefore, direct substitution produces:
0/0
The form 0/0 is called an indeterminate form. It tells us that direct substitution alone cannot determine the value of the limit. We must simplify the algebraic expression before evaluating it.
Step 2: Factorize the Numerator
The numerator of the given expression is:
2x + 6
Taking 2 as the common factor:
2x + 6 = 2(x + 3)
Therefore, the original limit becomes:
limx→−3 [2(x + 3)/(x + 3)]
Now the common factor x + 3 appears in both the numerator and denominator.
Step 3: Cancel the Common Factor
For x ≠ −3, the common factor x + 3 can be cancelled from the numerator and denominator:
2(x + 3)/(x + 3) = 2
Therefore:
limx→−3 (2x + 6)/(x + 3)
=
limx→−3 2
The limit of a constant is equal to the constant itself. Hence:
limx→−3 2 = 2
Therefore, the required value of the limit is:
2
Why Cancelling x + 3 Is Valid in a Limit
At x = −3, the original expression is not defined because its denominator becomes zero. However, a limit does not ask for the value of the function exactly at x = −3. Instead, it asks what value the function approaches when x becomes arbitrarily close to −3.
For every value of x close to −3 but not exactly equal to −3, the factor x + 3 is non-zero and can be cancelled. Thus:
(2x + 6)/(x + 3) = 2, for x ≠ −3
Since the expression has the constant value 2 at all nearby points, the function approaches 2 as x approaches −3. Therefore, the limit exists even though the original function is undefined at x = −3.
Understanding the Indeterminate Form 0/0
The expression 0/0 is called an indeterminate form because it does not provide enough information to determine a limit directly. Different functions can produce 0/0 after substitution but have completely different limiting values.
In the present problem, the indeterminate form occurs because both the numerator and denominator contain the common factor x + 3. Once this factor is removed, the true behavior of the function near x = −3 becomes clear.
The simplification is:
(2x + 6)/(x + 3)
=
2(x + 3)/(x + 3)
=
2
Therefore, the indeterminate form disappears after algebraic simplification, and the limit can be evaluated immediately.
Geometrical Interpretation of the Limit
The function:
f(x) = (2x + 6)/(x + 3)
simplifies to:
f(x) = 2, for x ≠ −3
Its graph is therefore the horizontal line y = 2 with a single missing point at x = −3. The missing point occurs because the original expression is undefined when x = −3.
This type of missing point is called a removable discontinuity. Although the function has no defined value at x = −3, the values of the function on both sides of −3 approach 2. Hence, the limit is 2.
Verification Using Left-Hand and Right-Hand Limits
A two-sided limit exists when the left-hand limit and right-hand limit are equal. After simplification, the expression is equal to 2 for every x near −3 except at x = −3 itself.
Therefore, as x approaches −3 from values smaller than −3:
limx→−3− (2x + 6)/(x + 3) = 2
Similarly, as x approaches −3 from values greater than −3:
limx→−3+ (2x + 6)/(x + 3) = 2
Since both one-sided limits are equal:
limx→−3 (2x + 6)/(x + 3) = 2
Alternative Solution Using L’Hôpital’s Rule
Since direct substitution gives the indeterminate form 0/0, L’Hôpital’s Rule can also be applied. According to this rule, the numerator and denominator can be differentiated separately when the required conditions are satisfied.
Differentiating the numerator:
d(2x + 6)/dx = 2
Differentiating the denominator:
d(x + 3)/dx = 1
Therefore:
limx→−3 (2x + 6)/(x + 3)
=
limx→−3 2/1
=
2
This method confirms the answer obtained by factorization. However, factorization is the simpler and more direct method for this particular problem.
Final Answer
The value of limx→−3 (2x + 6)/(x + 3) is 2.
