Textbook Question
Determine the following limits.
lim p→1 p^5 − 1 / p − 1
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1. Limits and Continuity
Finding Limits Algebraically
2:24 minutesProblem 51
Textbook Question
Determine the following limits.
lim x→1^− x/ ln x
Verified step by step guidance1
Identify the type of limit: This is a one-sided limit as \( x \to 1^- \), meaning we approach 1 from the left.
Substitute \( x = 1 \) into the expression \( \frac{x}{\ln x} \) to check if it results in an indeterminate form.
Recognize that as \( x \to 1^- \), \( \ln x \to \ln(1) = 0 \) and \( x \to 1 \), leading to the indeterminate form \( \frac{1}{0} \).
Apply L'Hôpital's Rule, which is used to evaluate limits of indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). Differentiate the numerator and the denominator separately.
Differentiate: The derivative of the numerator \( x \) is 1, and the derivative of the denominator \( \ln x \) is \( \frac{1}{x} \). Rewrite the limit as \( \lim_{x \to 1^-} \frac{1}{\frac{1}{x}} \).
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits
Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding the function's behavior near points of interest, including points of discontinuity or infinity. In this case, we are interested in the limit as x approaches 1 from the left, which requires analyzing the function's values as they get closer to this point.
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Natural Logarithm (ln)
The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is approximately 2.71828. It is a crucial function in calculus, particularly in limits and derivatives, as it has unique properties, such as being undefined for non-positive values. Understanding how ln(x) behaves as x approaches 1 is essential for evaluating the limit in the given question.
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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 the limit lim x→1^− x/ln x, both the numerator and denominator approach 0 as x approaches 1 from the left, creating an indeterminate form. Recognizing this allows us to apply techniques such as L'Hôpital's Rule to resolve the limit.
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