Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.
lim h→0 √16 + h − 4 / h

Verified step by step guidance

1. Recognize that the limit \( \lim_{h \to 0} \frac{\sqrt{16 + h} - 4}{h} \) is an indeterminate form of type \( \frac{0}{0} \).
2. To resolve the indeterminate form, multiply the numerator and the denominator by the conjugate of the numerator: \( \frac{\sqrt{16 + h} - 4}{h} \times \frac{\sqrt{16 + h} + 4}{\sqrt{16 + h} + 4} \).
3. Simplify the expression: the numerator becomes \((\sqrt{16 + h} - 4)(\sqrt{16 + h} + 4) = (16 + h) - 16 = h\).
4. The expression simplifies to \( \frac{h}{h(\sqrt{16 + h} + 4)} \), and the \( h \) terms cancel out, leaving \( \frac{1}{\sqrt{16 + h} + 4} \).
5. Evaluate the limit by substituting \( h = 0 \) into the simplified expression: \( \lim_{h \to 0} \frac{1}{\sqrt{16 + h} + 4} \).

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 evaluating expressions that may be undefined at those points. In this case, we are interested in the limit as h approaches 0.

Rationalizing the Numerator

Rationalizing the numerator is a technique used to simplify expressions involving square roots. By multiplying the numerator and denominator by the conjugate of the numerator, we can eliminate the square root, making it easier to evaluate the limit. This method is particularly useful when dealing with limits that result in 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. Recognizing these forms is essential, as they often require additional techniques, such as factoring, rationalizing, or applying L'Hôpital's Rule, to resolve and find the actual limit.