Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_x→ 1 ln x / (4x - x² - 3)

Verified step by step guidance

1. First, substitute x = 1 into the expression to check if the limit results in an indeterminate form. This will help us decide if l'Hôpital's Rule is applicable.
2. After substitution, observe that both the numerator ln(x) and the denominator (4x - x² - 3) evaluate to 0, indicating a 0/0 indeterminate form.
3. Since we have a 0/0 indeterminate form, we can apply l'Hôpital's Rule. This rule states that for limits of the form 0/0 or ∞/∞, the limit of the ratio of the derivatives of the numerator and the denominator can be taken instead.
4. Differentiate the numerator ln(x) with respect to x, which gives 1/x. Differentiate the denominator (4x - x² - 3) with respect to x, which gives 4 - 2x.
5. Now, evaluate the limit of the new expression (1/x) / (4 - 2x) as x approaches 1. Substitute x = 1 into this 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 context, evaluating the limit as x approaches 1 involves determining the behavior of the function ln(x) / (4x - x² - 3) near that point.

l'Hôpital's Rule

l'Hôpital's Rule is a method for evaluating limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if these forms occur, 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 simplifies the process of finding limits when direct substitution is not possible.

Natural Logarithm

The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is approximately 2.718. It is a crucial function in calculus, particularly in limits and derivatives, due to its unique properties, such as ln(1) = 0. Understanding how ln(x) behaves as x approaches 1 is vital for evaluating the given limit.