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→−95x
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1. Limits and Continuity
Finding Limits Algebraically
1:46 minutesProblem 2.29
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→3 −5x / √4x − 3
Verified step by step guidance1
Identify the limit expression: \( \lim_{{x \to 3}} \frac{{-5x}}{{\sqrt{{4x}} - 3}} \).
Substitute \( x = 3 \) into the expression to check if it results in an indeterminate form.
Calculate \( \sqrt{{4 \times 3}} - 3 \) to see if the denominator becomes zero.
If the expression is indeterminate, consider rationalizing the denominator or using L'Hôpital's Rule if applicable.
Evaluate the limit using the chosen method to simplify the expression.
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Video duration:
1mKey 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 specific points, which is crucial for evaluating continuity and differentiability. In this case, we are interested in the limit of the function as x approaches 3.
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Rational Functions
Rational functions are expressions formed by the ratio of two polynomials. They can exhibit different behaviors depending on the values of x, particularly at points where the denominator is zero. Understanding how to simplify and analyze these functions is essential for finding limits, especially when approaching points that may lead to indeterminate forms.
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Indeterminate Forms
Indeterminate forms occur when evaluating limits leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit's value. Recognizing these forms is crucial, as they often require additional techniques, such as L'Hôpital's Rule or algebraic manipulation, to resolve and find the actual limit.
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