Textbook Question
Find the limits in Exercises 49–52. Write ∞ or −∞ where appropriate.
lim θ→0 (2 − cot θ)
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1. Limits and Continuity
Finding Limits Algebraically
2:41 minutesProblem 56b
Textbook Question
Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.
lim (x² − 1) / (2x + 4) as
b. x→−2⁻
Verified step by step guidance1
Identify the type of limit: This is a one-sided limit as x approaches -2 from the left (x → -2⁻).
Substitute x = -2 into the expression (x² − 1) / (2x + 4) to check if it results in an indeterminate form. Calculate the numerator: (-2)² - 1 = 4 - 1 = 3. Calculate the denominator: 2(-2) + 4 = -4 + 4 = 0. The expression is of the form 3/0, indicating a potential vertical asymptote.
Determine the sign of the expression as x approaches -2 from the left. Choose a test value slightly less than -2, such as x = -2.1. Calculate the denominator: 2(-2.1) + 4 = -4.2 + 4 = -0.2, which is negative.
Since the numerator is positive (3) and the denominator is negative as x approaches -2 from the left, the overall expression approaches negative infinity.
Conclude that the limit of (x² − 1) / (2x + 4) as x approaches -2 from the left is -∞.
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Video duration:
2mKey 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, including points of discontinuity or infinity. Evaluating limits is essential for determining the continuity of functions and for finding derivatives.
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One-Sided Limits
One-Sided Limits
One-sided limits refer to the value that a function approaches as the input approaches a specific point from one side only, either the left (denoted as x→a⁻) or the right (denoted as x→a⁺). This concept is crucial when dealing with functions that may behave differently from each side of a point, particularly at points of discontinuity.
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Rational Functions
Rational functions are functions that can be expressed as the ratio of two polynomials. Understanding their limits often involves analyzing the behavior of the numerator and denominator as the variable approaches a certain value. In this case, identifying how the function behaves as x approaches -2 is key to determining the limit.
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