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Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 2 - Limits and ContinuityProblem 2.3.43
Chapter 2, Problem 2.3.43

Using the Formal Definition


Prove the limit statements in Exercises 37–50.


lim x→1 1/x = 1

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1
Understand the formal definition of a limit: For a function f(x) and a limit L, the statement lim x→c f(x) = L means that for every ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε.
Identify the components of the limit statement: Here, f(x) = 1/x, L = 1, and c = 1. We need to show that for every ε > 0, there exists a δ > 0 such that if 0 < |x - 1| < δ, then |1/x - 1| < ε.
Express the condition |1/x - 1| < ε in terms of x: Start by simplifying the expression |1/x - 1|. This can be rewritten as |(1 - x)/x|. We want this to be less than ε.
Find a suitable δ: To ensure |(1 - x)/x| < ε, we can manipulate the inequality. Notice that |(1 - x)/x| = |1 - x|/|x|. We need to find δ such that |1 - x| < ε|x|. Assume |x - 1| < 1, which implies 0 < x < 2, and thus 1/2 < 1/x < 1. This helps in bounding |x|.
Conclude the proof: Choose δ = min(1, ε/2). This choice ensures that if 0 < |x - 1| < δ, then |1/x - 1| < ε, satisfying the formal definition of the limit. Therefore, lim x→1 1/x = 1 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) to have a limit L as x approaches a value a, for every ε > 0, there exists a δ > 0 such that if 0 < |x - a| < δ, then |f(x) - L| < ε. This definition is crucial for rigorously proving limit statements in calculus.
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Epsilon-Delta Proof

An epsilon-delta proof is a method used to demonstrate the validity of a limit using the formal definition. It involves selecting an appropriate δ for a given ε to show that the function's output can be made arbitrarily close to the limit by choosing inputs sufficiently close to the point of interest.
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Continuity and Limits

A function is continuous at a point if the limit as x approaches that point equals the function's value at that point. Understanding continuity helps in analyzing limits, as continuous functions behave predictably near their points, making it easier to apply the epsilon-delta definition.
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Related Practice
Textbook Question

Limits of Average Rates of Change


Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form limh→0 (f(x+h) − f(x)) / h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f.


f(x) = 1/x, x = -2

Textbook Question

Limits of quotients


Find the limits in Exercises 23–42.


limx→−1 (√(x² + 8) − 3) / (x + 1)

Textbook Question

At what points are the functions in Exercises 13–30 continuous?


g(x) = { (x² − x – 6)/(x – 3), x ≠ 3

5, x = 3


Textbook Question

At what points are the functions in Exercises 13–30 continuous?

y = 1/(x – 2) – 3x

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)

Textbook Question

Never-zero continuous functions Is it true that a continuous function that is never zero on an interval never changes sign on that interval? Give reasons for your answer.