Textbook Question
Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.
lim x→2 (5x−6)^3/2
1. Limits and Continuity
Finding Limits Algebraically
3:14 minutesProblem 41
Textbook Question
Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.
lim x→9 √x − 3 / x − 9
Verified step by step guidance1
Recognize that the limit \( \lim_{x \to 9} \frac{\sqrt{x} - 3}{x - 9} \) is an indeterminate form \( \frac{0}{0} \).
To resolve the indeterminate form, use algebraic manipulation. Multiply the numerator and the denominator by the conjugate of the numerator: \( \sqrt{x} + 3 \).
This gives: \( \lim_{x \to 9} \frac{(\sqrt{x} - 3)(\sqrt{x} + 3)}{(x - 9)(\sqrt{x} + 3)} \).
Simplify the expression: the numerator becomes \( x - 9 \) because \((\sqrt{x} - 3)(\sqrt{x} + 3) = x - 9\).
Cancel \( x - 9 \) from the numerator and denominator, resulting in \( \lim_{x \to 9} \frac{1}{\sqrt{x} + 3} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits
A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points where they may not be defined. In this case, we are interested in the limit of a function as x approaches 9.
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Indeterminate Forms
Indeterminate forms occur in calculus when direct substitution in a limit leads to an ambiguous result, such as 0/0. In the given limit, substituting x = 9 results in this form, necessitating further analysis, such as algebraic manipulation or applying L'Hôpital's Rule to resolve the limit.
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Rationalizing the Numerator
Rationalizing the numerator is a technique used to simplify expressions involving square roots. By multiplying the numerator and denominator by the conjugate of the numerator, we can eliminate the square root and simplify the limit calculation. This method is particularly useful when dealing with limits that yield indeterminate forms.
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