lim_x→3 (x² − 9)/(x² − 4x + 3)

is ______ (rounded off to the nearest integer).

Evaluate the Limit of (x² − 9)/(x² − 4x + 3) as x Approaches 3

Understanding the Given Limit Problem

This question asks us to evaluate the limit of a rational algebraic function as x approaches 3. The given expression contains quadratic polynomials in both the numerator and denominator:

lim_x→3 (x² − 9)/(x² − 4x + 3)

When evaluating a limit of a rational function, the first step is usually direct substitution. If direct substitution produces a definite real number, the limit can be obtained immediately. However, if substitution produces the indeterminate form 0/0, the expression must be simplified before the limit can be evaluated.

In this problem, direct substitution gives 0/0. This does not mean that the answer is zero or that the limit does not exist. Instead, it indicates that the numerator and denominator contain a common factor that becomes zero at x = 3.

Step-by-Step Solution

Step 1: Apply Direct Substitution

We begin by substituting x = 3 into the numerator:

x² − 9 = 3² − 9

Therefore:

9 − 9 = 0

Now substitute x = 3 into the denominator:

x² − 4x + 3 = 3² − 4(3) + 3

Thus:

9 − 12 + 3 = 0

Therefore, direct substitution gives:

0/0

The form 0/0 is an indeterminate form. It indicates that further algebraic simplification is required.

Step 2: Factorize the Numerator

The numerator is:

x² − 9

This is a difference of two squares because:

x² − 9 = x² − 3²

Using the identity:

a² − b² = (a − b)(a + b)

we obtain:

x² − 9 = (x − 3)(x + 3)

Step 3: Factorize the Denominator

The denominator is:

x² − 4x + 3

To factorize this quadratic expression, we need two numbers whose product is 3 and whose sum is −4. These numbers are −1 and −3.

Therefore:

x² − 4x + 3 = (x − 1)(x − 3)

The original limit can now be written as:

lim_x→3 [(x − 3)(x + 3)]/[(x − 1)(x − 3)]

Step 4: Cancel the Common Factor

The factor x − 3 appears in both the numerator and denominator. For values of x close to 3 but not exactly equal to 3, this common factor can be cancelled.

Therefore:

[(x − 3)(x + 3)]/[(x − 1)(x − 3)] = (x + 3)/(x − 1)

Hence, the limit becomes:

lim_x→3 (x + 3)/(x − 1)

Step 5: Substitute x = 3 into the Simplified Expression

After cancellation, the simplified expression is defined at x = 3. Therefore, we can now use direct substitution:

(3 + 3)/(3 − 1)

Thus:

6/2 = 3

Therefore:

lim_x→3 (x² − 9)/(x² − 4x + 3) = 3

Step 6: Round the Answer to the Nearest Integer

The calculated value of the limit is exactly 3. Since 3 is already an integer, no further rounding is required.

Therefore:

Answer = 3

Complete Simplification in a Single Calculation

The entire solution can be written compactly as:

lim_x→3 (x² − 9)/(x² − 4x + 3)

= lim_x→3 [(x − 3)(x + 3)]/[(x − 3)(x − 1)]

= lim_x→3 (x + 3)/(x − 1)

= (3 + 3)/(3 − 1)

= 6/2

= 3

Why the Indeterminate Form 0/0 Does Not Mean the Answer Is Zero

When direct substitution gives 0/0, the result is called an indeterminate form. It does not represent a numerical value. Therefore, we cannot conclude that the limit is zero.

In this problem, both the numerator and denominator become zero because they share the factor x − 3. The factorization reveals that the zero in the numerator and the zero in the denominator arise from the same common factor.

After cancelling this factor, the remaining expression approaches the definite value 3. Therefore, the original 0/0 form simply indicates that algebraic simplification is necessary.

Why Cancelling the Factor x − 3 Is Valid

At x = 3, the original expression is undefined because its denominator is zero. However, a limit is concerned with values of x approaching 3, not necessarily with the value of the function exactly at x = 3.

For every value of x sufficiently close to 3 but not equal to 3, the factor x − 3 is non-zero and can be cancelled. Therefore, near x = 3:

(x² − 9)/(x² − 4x + 3) = (x + 3)/(x − 1)

The simplified function approaches 3 as x approaches 3, so the original function has the same limit.

Alternative Solution Using L’Hôpital’s Rule

Since direct substitution gives the indeterminate form 0/0, the limit can also be evaluated using L’Hôpital’s Rule.

The numerator is:

x² − 9

Its derivative is:

2x

The denominator is:

x² − 4x + 3

Its derivative is:

2x − 4

Therefore:

lim_x→3 (x² − 9)/(x² − 4x + 3)
=
lim_x→3 2x/(2x − 4)

Substituting x = 3:

2(3)/[2(3) − 4]

= 6/(6 − 4)

= 6/2

= 3

This confirms the result obtained by factorization. For this particular problem, factorization is the simpler and more direct method.

Geometrical Interpretation of the Limit

The original rational function is undefined at x = 3 because both its numerator and denominator become zero. However, after cancelling the common factor x − 3, the function behaves like:

(x + 3)/(x − 1)

for all nearby values of x except x = 3 itself. The simplified expression has the value 3 at x = 3.

Therefore, the graph of the original function has a removable discontinuity or a hole at the point corresponding to x = 3 and y = 3. Even though the original function is undefined exactly at that point, its values approach 3 from both sides.

Final Answer

The value of lim_x→3 (x² − 9)/(x² − 4x + 3) is 3.

Answer: 3