Finding Limits
In Exercises 9–24, find the limit or explain why it does not exist.




lim x→a (x² ― a²)/(x⁴ ― a⁴)

Verified step by step guidance

1. First, recognize that the expression (x² - a²)/(x⁴ - a⁴) is a rational function. To find the limit as x approaches a, we need to simplify the expression, especially since direct substitution would lead to an indeterminate form 0/0.
2. Notice that both the numerator and the denominator are differences of squares. The numerator x² - a² can be factored as (x - a)(x + a). Similarly, the denominator x⁴ - a⁴ can be factored using the difference of squares twice: first as (x² - a²)(x² + a²), and then further factor x² - a² as (x - a)(x + a).
3. After factoring, the expression becomes ((x - a)(x + a))/((x - a)(x + a)(x² + a²)). Cancel the common factors (x - a)(x + a) from the numerator and the denominator, provided x ≠ a.
4. The simplified expression is now 1/(x² + a²). This expression is no longer indeterminate as x approaches a.
5. Finally, evaluate the limit by substituting x = a into the simplified expression, which is now straightforward since it no longer results in an indeterminate form.

Key Concepts

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near specific points, which is crucial for analyzing continuity, derivatives, and integrals. In this case, we are interested in the limit of a rational function as x approaches a.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. They can exhibit unique behaviors, such as asymptotes and discontinuities, depending on the values of x. In the given limit problem, both the numerator and denominator are polynomials, and their degrees will influence the limit's existence and value as x approaches a.

Factoring and Simplifying

Factoring and simplifying expressions is a key technique in calculus for evaluating limits, especially when direct substitution leads to indeterminate forms like 0/0. By factoring the numerator and denominator, we can often cancel common terms, making it easier to find the limit as x approaches a. This process is essential for resolving the limit in the given exercise.