Evaluating the Limit of (x^2 - 4)/(x - 2) as x Approaches 2

In mathematics, limits are often encountered in scenarios where direct substitution leads to indeterminate forms. One such form is 0/0. In this article, we will explore how to evaluate the limit of the expression mathfrac{x^2 - 4}{x - 2}/math as x approaches 2. We will discuss the steps to resolve indeterminate forms and use the algebraic factoring method and L'H?pital's rule to find the solution.

Understanding the Expression

Consider the expression mathfrac{x^2 - 4}{x - 2}/math. When we substitute x 2, we get:

mathfrac{2^2 - 4}{2 - 2} frac{4 - 4}{0} frac{0}{0}/math

This results in an indeterminate form 0/0. Indeterminate forms do not have a fixed value and need to be resolved to find the actual limit.

Resolving the Indeterminate Form

Algebraic Factoring

To resolve the indeterminate form, we can factor the numerator:

mathx^2 - 4 (x 2)(x - 2)/math

Substituting this factorization into the original expression, we get:

mathfrac{(x 2)(x - 2)}{x - 2}/math

When x ≠ 2, we can cancel out the common factor of x - 2 in the numerator and denominator:

mathfrac{(x 2)(x - 2)}{x - 2} x 2/math

Now, we can substitute x 2 into the simplified expression:

mathx 2 2 2 4/math

Therefore, the limit of the expression as x approaches 2 is 4.

L'H?pital's Rule

Alternatively, for those familiar with L'H?pital's rule, which states that if the limit of the ratio of two functions as x approaches a certain value is the indeterminate form 0/0, then this limit is equal to the limit of the ratio of their derivatives:

mathlim_{x to a} frac{f(x)}{g(x)} lim_{x to a} frac{f'(x)}{g'(x)}/math

Let's apply L'H?pital's rule to our expression:

mathlim_{x to 2} frac{x^2 - 4}{x - 2} lim_{x to 2} frac{2x}{1} 2(2) 4/math

The derivative of the numerator x^2 - 4 is 2x, and the derivative of the denominator x - 2 is 1. Substituting x 2 into the derivative form, we again get 4.

Conclusion

Regardless of the method used, whether it involves algebraic factoring or L'H?pital's rule, the limit of the expression mathfrac{x^2 - 4}{x - 2}/math as x approaches 2 is 4. This result demonstrates the importance of understanding and resolving indeterminate forms in mathematical calculations.

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