Textbook Question
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 0 (1 - cos 3x) / 8x²
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4. Applications of Derivatives
Differentials
5:57 minutesProblem 89
Textbook Question
More limits Evaluate the following limits.
lim_x→1 (x ln x - x + 1) / (xln²x)
Verified step by step guidance1
First, identify the form of the limit as x approaches 1. Substitute x = 1 into the expression (x ln x - x + 1) / (x ln²x) to check if it results in an indeterminate form like 0/0.
Since substituting x = 1 results in an indeterminate form, apply L'Hôpital's Rule, which is used to evaluate limits of indeterminate forms. This rule states that if the limit of f(x)/g(x) as x approaches a value results in 0/0 or ∞/∞, then it can be evaluated as the limit of f'(x)/g'(x).
Differentiate the numerator f(x) = x ln x - x + 1 with respect to x. Use the product rule for the term x ln x, which gives the derivative as ln x + 1. The derivative of -x is -1, and the derivative of the constant 1 is 0.
Differentiate the denominator g(x) = x ln²x with respect to x. Again, use the product rule. The derivative of x is 1, and the derivative of ln²x is 2 ln x * (1/x) by the chain rule. Combine these to get the derivative of the denominator.
Substitute the derivatives back into the limit expression and evaluate the new limit as x approaches 1. If necessary, apply L'Hôpital's Rule again if the result is still an indeterminate form.
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Video duration:
5mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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 analyzing the behavior of the function near that point, which may require techniques like substitution or L'Hôpital's rule if the limit results in an indeterminate form.
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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 crucial for applying methods such as L'Hôpital's rule, which allows for differentiation of the numerator and denominator to resolve the limit. In the given limit, substituting x = 1 results in an indeterminate form, necessitating further analysis.
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L'Hôpital's Rule
L'Hôpital's Rule is a powerful tool in calculus used to evaluate limits that result in indeterminate forms. It states that if the limit of f(x)/g(x) as x approaches a point yields 0/0 or ∞/∞, then the limit can be found by taking the derivative of the numerator and the derivative of the denominator. This rule simplifies the evaluation process, making it easier to find the limit of complex functions like the one presented in the question.
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