lim_x→2 [(2x² + 2x − 12)/(x² − 4)]

is __________. (rounded off to 1 decimal)

Limit of a Rational Function as x Approaches 2 – Detailed Step-by-Step Solution

Understanding the Given Limit Problem

In this question, we need to evaluate the limit of a rational function as x approaches 2. The given function is a ratio of two polynomial expressions. For such limit problems, the first and most important step is direct substitution. Direct substitution helps us determine whether the function can be evaluated immediately or whether it produces an indeterminate form that requires algebraic simplification.

The given limit is:

lim_x→2
[(2x² + 2x − 12)/(x² − 4)]

Step-by-Step Solution of the Limit

Step 1: Apply Direct Substitution

We first substitute x = 2 into the numerator and denominator. This is always the correct starting point when evaluating an algebraic limit.

For the numerator:

2(2)² + 2(2) − 12

= 2(4) + 4 − 12

= 8 + 4 − 12

= 0

For the denominator:

(2)² − 4

= 4 − 4

= 0

Therefore, direct substitution gives 0/0. This is called an indeterminate form. It does not mean that the limit is zero, and it also does not mean that the limit does not exist. Instead, the appearance of 0/0 tells us that the original expression must be simplified before the limit can be evaluated.

Step 2: Factorize the Numerator

The numerator of the function is:

2x² + 2x − 12

First, take 2 as a common factor:

2x² + 2x − 12

= 2(x² + x − 6)

Now factorize x² + x − 6. We need two numbers whose product is −6 and whose sum is +1. These numbers are +3 and −2.

x² + x − 6

= (x + 3)(x − 2)

Therefore, the numerator becomes:

2(x + 3)(x − 2)

Step 3: Factorize the Denominator

The denominator is:

x² − 4

This expression is a difference of two squares. Using the identity a² − b² = (a − b)(a + b), we get:

x² − 4

= x² − 2²

= (x − 2)(x + 2)

Step 4: Substitute the Factorized Expressions

After factorizing both the numerator and denominator, the original limit becomes:

lim_x→2
[2(x + 3)(x − 2)/((x − 2)(x + 2))]

The factor (x − 2) is common to both the numerator and denominator. Since a limit considers values of x approaching 2 rather than requiring x to be exactly equal to 2 during the simplification process, this common factor can be cancelled.

= lim_x→2 [2(x + 3)/(x + 2)]

Step 5: Evaluate the Simplified Limit

The simplified expression no longer produces the indeterminate form 0/0. Therefore, we can now substitute x = 2 directly:

lim_x→2 [2(x + 3)/(x + 2)]

= 2(2 + 3)/(2 + 2)

= 2(5)/4

= 10/4

= 2.5

Why the 0/0 Form Does Not Give the Final Answer

A key concept in solving limits is understanding the meaning of the 0/0 indeterminate form. When both the numerator and denominator become zero after direct substitution, the result cannot be interpreted as ordinary division. The expression often contains a common factor responsible for making both parts zero. In this problem, that common factor is (x − 2).

After factorization and cancellation of the common factor, the true behavior of the function near x = 2 becomes visible. The simplified function approaches 2.5, even though the original expression is undefined exactly at x = 2. This distinction between the value of a function at a point and the limit of a function near that point is one of the most fundamental ideas in calculus.

Final Answer

lim_x→2
[(2x² + 2x − 12)/(x² − 4)]
= 2.5

The calculated value is already expressed to one decimal place. Therefore, the required answer is:

Correct Answer: 2.5