Use an appropriate limit definition to prove the following limits.

lim x→ 5x^2 − 25 / x − 5=10

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Identify the limit expression: \( \lim_{{x \to 5}} \frac{{x^2 - 25}}{{x - 5}} \).

Recognize that the expression \( x^2 - 25 \) can be factored as \((x - 5)(x + 5)\).

Rewrite the limit expression using the factorization: \( \lim_{{x \to 5}} \frac{{(x - 5)(x + 5)}}{{x - 5}} \).

Cancel the common factor \((x - 5)\) in the numerator and denominator, simplifying the expression to \( \lim_{{x \to 5}} (x + 5) \).

Evaluate the limit by direct substitution: substitute \(x = 5\) into the simplified expression \(x + 5\).

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Limit Definition

The limit definition in calculus refers to the formal approach to finding the limit of a function as it approaches a certain point. It involves evaluating the behavior of the function as the input values get arbitrarily close to a specified value, often denoted as 'a'. This concept is foundational for understanding continuity, derivatives, and integrals.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial expression into simpler components, or factors, that can be multiplied together to yield the original polynomial. In the context of limits, factoring can help simplify expressions that yield indeterminate forms, such as 0/0, allowing for easier evaluation of the limit.

Indeterminate Forms

Indeterminate forms occur in calculus when evaluating limits leads to expressions that do not provide clear information about the limit's value, such as 0/0 or ∞/∞. Recognizing these forms is crucial, as they often require additional techniques, like factoring or L'Hôpital's Rule, to resolve and find the actual limit.

Use an appropriate limit definition to prove the following limits.lim x→ 5x^2 − 25 / x − 5=10