Determine the following limits.
lim h→0 (h + 6)^2 + (h + 6) − 42 / h

Verified step by step guidance

1. Step 1: Identify the expression whose limit we need to find. The expression is \( \frac{(h + 6)^2 + (h + 6) - 42}{h} \).
2. Step 2: Simplify the numerator. Expand \((h + 6)^2\) to get \(h^2 + 12h + 36\).
3. Step 3: Combine the terms in the numerator. The expression becomes \(h^2 + 12h + 36 + h + 6 - 42\).
4. Step 4: Simplify the combined terms. This results in \(h^2 + 13h\).
5. Step 5: Factor the numerator. The expression \(h^2 + 13h\) can be factored as \(h(h + 13)\).

Key Concepts

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are essential for understanding continuity, derivatives, and integrals. In this question, evaluating the limit as h approaches 0 helps determine the behavior of the function near that point.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to make calculations easier. In the context of limits, it often includes factoring, expanding, or combining like terms to eliminate indeterminate forms such as 0/0. This skill is crucial for simplifying the expression before applying limit laws.

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms like 0/0 or ∞/∞. It states that if such a form occurs, the limit of the ratio of two functions can be found by taking the derivative of the numerator and the derivative of the denominator. This rule can simplify the process of finding limits in complex expressions.