Evaluate each limit and justify your answer. 
lim x→4 √x^3−2x^2−8x / x−4

Verified step by step guidance

1. Identify the type of limit: This is a limit of the form \( \frac{0}{0} \) as both the numerator and denominator approach zero when \( x \to 4 \).
2. Apply L'Hôpital's Rule: Since the limit is of the indeterminate form \( \frac{0}{0} \), we can apply L'Hôpital's Rule, which involves differentiating the numerator and the denominator separately.
3. Differentiate the numerator: Find the derivative of \( \sqrt{x^3 - 2x^2 - 8x} \) with respect to \( x \). Use the chain rule and power rule for differentiation.
4. Differentiate the denominator: The derivative of \( x - 4 \) with respect to \( x \) is simply 1.
5. Evaluate the new limit: Substitute \( x = 4 \) into the new expression obtained after applying L'Hôpital's Rule and simplify to find the limit.

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 points of interest, including points where they may not be defined. Evaluating limits often involves techniques such as substitution, factoring, or applying L'Hôpital's rule when dealing with indeterminate forms.

Indeterminate Forms

Indeterminate forms occur when direct substitution in a limit leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit's value. In such cases, further analysis is required, often involving algebraic manipulation or L'Hôpital's rule, which allows for differentiation of the numerator and denominator to resolve the limit.

Factoring and Simplification

Factoring and simplification are techniques used to rewrite expressions in a more manageable form, especially when evaluating limits. By factoring out common terms or simplifying complex expressions, one can often eliminate indeterminate forms and make it easier to compute the limit. This process is crucial in finding the limit of rational functions or polynomials.