Evaluate each limit and justify your answer. 
lim x→5 ln 6(√x^2−16−3) / 5x−25

Verified step by step guidance

1. Step 1: Recognize that the limit involves a rational function where both the numerator and the denominator approach 0 as x approaches 5. This is an indeterminate form of type 0/0, which suggests the use of L'Hôpital's Rule.
2. Step 2: Apply L'Hôpital's Rule, which states that if the limit of f(x)/g(x) as x approaches a value results in an indeterminate form 0/0 or ∞/∞, then the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately.
3. Step 3: Differentiate the numerator ln(6(√x^2−16)−3) with respect to x. Use the chain rule and the derivative of the natural logarithm function. The derivative of ln(u) is 1/u * du/dx, where u = 6(√x^2−16)−3.
4. Step 4: Differentiate the denominator 5x−25 with respect to x. The derivative is straightforward since it is a linear function, resulting in 5.
5. Step 5: Evaluate the new limit using the derivatives obtained in Steps 3 and 4. Substitute x = 5 into the resulting expression to find the limit.

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 x approaches 5 involves determining the behavior of the function near that point, which may require algebraic manipulation or applying limit laws.

Natural Logarithm

The natural logarithm, denoted as ln, is the logarithm to the base e (approximately 2.718). It is a crucial function in calculus, particularly in problems involving growth rates and exponential functions. In the given limit, the natural logarithm of an expression is involved, which may affect the limit's evaluation, especially if the argument approaches zero or infinity.

Indeterminate Forms

Indeterminate forms occur in calculus when evaluating limits leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit's value. In this case, the limit may require techniques such as L'Hôpital's Rule or algebraic simplification to resolve. Recognizing and addressing indeterminate forms is crucial for correctly evaluating the limit presented in the question.