The value of \(\mathop {\lim }\limits_{x \to 2} \frac{{{x^2} - 4}}{{3x - 6}}\) is:
Limit Expression Analysis
To determine the value of the limit \( \mathop {\lim }\limits_{x \to 2} \frac{{{x^2} - 4}}{{3x - 6}} \), we begin by substituting the value \(x=2\) directly into the given expression. This initial step helps us identify the form of the limit.
- Numerator: When \(x=2\), the numerator becomes \( x^2 - 4 = (2)^2 - 4 = 4 - 4 = 0 \).
- Denominator: When \(x=2\), the denominator becomes \( 3x - 6 = 3(2) - 6 = 6 - 6 = 0 \).
Since both the numerator and the denominator evaluate to zero, the limit is in the indeterminate form \( \frac{0}{0} \). To resolve this, we need to simplify the rational expression, typically by factorizing the components.
Factorizing Limit Components
To simplify the expression \( \frac{{{x^2} - 4}}{{3x - 6}} \), we will factorize the numerator and the denominator separately:
- Numerator factorization: The numerator, \( x^2 - 4 \), is a difference of squares. It follows the algebraic identity \( a^2 - b^2 = (a-b)(a+b) \). Here, \( a=x \) and \( b=2 \).
Thus, \( x^2 - 4 \) can be factored as \( (x-2)(x+2) \). - Denominator factorization: The denominator, \( 3x - 6 \), has a common factor of 3.
Thus, \( 3x - 6 \) can be factored as \( 3(x-2) \).
Simplifying the Rational Expression
Now, substitute the factored forms back into the original limit expression:
\( \mathop {\lim }\limits_{x \to 2} \frac{{{x^2} - 4}}{{3x - 6}} = \mathop {\lim }\limits_{x \to 2} \frac{{(x-2)(x+2)}}{{3(x-2)}} \)
As \( x \) approaches 2 (but is not exactly 2), the term \( (x-2) \) is not equal to zero. This allows us to cancel out the common factor \( (x-2) \) from both the numerator and the denominator, simplifying the expression significantly.
\( \mathop {\lim }\limits_{x \to 2} \frac{{(x-2)(x+2)}}{{3(x-2)}} = \mathop {\lim }\limits_{x \to 2} \frac{{x+2}}{{3}} \)
Final Limit Value Calculation
With the expression simplified to \( \frac{{x+2}}{{3}} \), we can now safely substitute \( x=2 \) into it to find the value of the limit:
\( \mathop {\lim }\limits_{x \to 2} \frac{{x+2}}{{3}} = \frac{{2+2}}{{3}} = \frac{{4}}{{3}} \)
Therefore, the calculated value of the limit \( \mathop {\lim }\limits_{x \to 2} \frac{{{x^2} - 4}}{{3x - 6}} \) is \( \frac{4}{3} \).
| Step Number | Description of Step | Mathematical Representation |
|---|---|---|
| 1 | Initial substitution (identifying the indeterminate form). | \( \frac{(2)^2 - 4}{3(2) - 6} = \frac{0}{0} \) |
| 2 | Factorization of numerator and denominator. | Numerator: \( (x-2)(x+2) \) Denominator: \( 3(x-2) \) |
| 3 | Cancellation of common factors and simplification. | \( \frac{x+2}{3} \) |
| 4 | Direct substitution into the simplified expression. | \( \frac{2+2}{3} = \frac{4}{3} \) |
Important Questions from Limits
The limit of the function f (x, y) = x + y - 6 at x = 1; y = 2 is ?
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