Textbook Question
Domains and Asymptotes
Determine the domain of each function in Exercises 69–72. Then use various limits to find the asymptotes.
y = (√(x² + 4)) / x
1. Limits and Continuity
Introduction to Limits
3:51 minutesProblem 2.6.93
Textbook Question
Use formal definitions to prove the limit statements in Exercises 93–96.
lim x → 0 (−1 / x²) = −∞
Verified step by step guidance1
Understand the formal definition of a limit approaching negative infinity: For every positive number M, there exists a positive number δ such that for all x, 0 < |x| < δ implies f(x) < -M.
Identify the function f(x) = -1/x² and the limit statement lim x → 0 (−1 / x²) = −∞.
Given any positive number M, we need to find a δ > 0 such that for all x, 0 < |x| < δ, the inequality -1/x² < -M holds.
Rearrange the inequality -1/x² < -M to 1/x² > M, which implies x² < 1/M. Therefore, |x| < 1/√M.
Choose δ = 1/√M. Then, for all x such that 0 < |x| < δ, it follows that 1/x² > M, which satisfies the condition for the limit to be -∞.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limit Definition
The formal definition of a limit involves showing that for every ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε. In this case, the limit is approaching negative infinity, which requires a modified approach to demonstrate that f(x) becomes arbitrarily large negative as x approaches 0.
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Behavior of Rational Functions
Understanding the behavior of rational functions, particularly as x approaches a value where the denominator tends to zero, is crucial. For the function −1/x², as x approaches 0, the denominator becomes very small, causing the function value to grow negatively without bound, leading to a limit of negative infinity.
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Infinity in Limits
When dealing with limits that approach infinity, the concept of infinity in calculus is used to describe unbounded behavior. A limit approaching negative infinity means the function values decrease without bound as x approaches the specified point. This requires demonstrating that for any large negative number, the function can be made smaller than that number by choosing x sufficiently close to the limit point.
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