4. Applications of Derivatives
Differentials
Problem 21
Textbook Question
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 guidance1
Identify the limit to evaluate: lim_{x→1} \frac{\ln x}{4x - x^2 - 3}. Substitute x = 1 to check if it results in an indeterminate form.
Calculate the numerator and denominator at x = 1: \ln(1) = 0 and 4(1) - (1)^2 - 3 = 0, confirming the indeterminate form 0/0.
Apply l'Hôpital's Rule, which states that if you have an indeterminate form 0/0, you can take the derivative of the numerator and the derivative of the denominator.
Differentiate the numerator: the derivative of \ln x is \frac{1}{x}. Differentiate the denominator: the derivative of (4x - x^2 - 3) is 4 - 2x.
Rewrite the limit using the derivatives: lim_{x→1} \frac{\frac{1}{x}}{4 - 2x} and substitute x = 1 to evaluate the limit.
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