Textbook Question
Finding Limits of Differences When x → ±∞
Find the limits in Exercises 84–90. (Hint: Try multiplying and dividing by the conjugate.)
lim x → −∞ (√(x² + 3) + x)
1. Limits and Continuity
Introduction to Limits
4:13 minutesProblem 2.6.96
Textbook Question
Use formal definitions to prove the limit statements in Exercises 93–96.
lim x → −5 (1 / (x + 5)²) = ∞
Verified step by step guidance1
To prove that \( \lim_{x \to -5} \frac{1}{(x + 5)^2} = \infty \), we need to use the formal definition of a limit approaching infinity. This means for every positive number \( M \), there exists a \( \delta > 0 \) such that if \( 0 < |x + 5| < \delta \), then \( \frac{1}{(x + 5)^2} > M \).
Start by considering the expression \( \frac{1}{(x + 5)^2} > M \). This inequality can be rewritten as \( (x + 5)^2 < \frac{1}{M} \).
To find \( \delta \), solve the inequality \( |x + 5| < \sqrt{\frac{1}{M}} \). This gives us the condition for \( \delta \) in terms of \( M \).
Set \( \delta = \sqrt{\frac{1}{M}} \). This choice of \( \delta \) ensures that whenever \( 0 < |x + 5| < \delta \), the inequality \( \frac{1}{(x + 5)^2} > M \) holds true.
Conclude that for every \( M > 0 \), there exists a \( \delta = \sqrt{\frac{1}{M}} \) such that if \( 0 < |x + 5| < \delta \), then \( \frac{1}{(x + 5)^2} > M \). Therefore, \( \lim_{x \to -5} \frac{1}{(x + 5)^2} = \infty \) is proven using the formal definition.
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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 states that for a function f(x), the limit as x approaches a value a is L if, for every ε > 0, there exists a δ > 0 such that whenever 0 < |x - a| < δ, it follows that |f(x) - L| < ε. This definition is crucial for proving limit statements rigorously.
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Behavior of Functions Near Asymptotes
Understanding how functions behave near vertical asymptotes is essential for limit proofs. In this case, as x approaches -5, the denominator (x + 5) approaches zero, causing the function (1 / (x + 5)²) to increase without bound, leading to the conclusion that the limit is infinity.
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Epsilon-Delta Proofs
Epsilon-delta proofs are a method used to rigorously establish limits. In this context, one must show that for any large number M (representing infinity), there exists a corresponding δ such that if x is within δ of -5 (but not equal to -5), then (1 / (x + 5)²) exceeds M, confirming the limit statement.
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