Problem 31 - Calculus Early Transcendentals: Pearson New International Edition

Question

Which of the following are equivalent to the definition of limit? (a) For some \(\varepsilon>0\) and every \(\delta>0,0<|x-c|<\delta \Rightarrow\) \(|f(x)-L|<\varepsilon\) (b) For every \(\delta>0\), there is a corresponding \(\varepsilon>0\) such that $$ 0<|x-c|<\varepsilon \Rightarrow|f(x)-L|<\delta $$ (c) For every positive integer \(N\), there is a corresponding positive integer \(M\) such that \(0<|x-c|<1 / M \Rightarrow|f(x)-L|\) \(<1 / N\) (d) For every \(\varepsilon>0\), there is a corresponding \(\delta>0\) such that \(0<|x-c|<\delta\) and \(|f(x)-L|<\varepsilon\) for some \(x\)

Step-by-Step Solution

Answer

Only option (c) is equivalent to the limit definition.

Step 1: Understanding the Real Definition of Limit

The definition of limit \(\lim_{{x \to c}} f(x) = L\) in \(\varepsilon-\delta\) terms is: For every \(\varepsilon > 0\), there exists a \(\delta > 0\) such that whenever \(0 < |x-c| < \delta\), it follows that \(|f(x)-L| < \varepsilon\).

Step 2: Analyzing Option A

Option (a) states: For some \(\varepsilon>0\) and every \(\delta>0\), \(0<|x-c|<\delta \Rightarrow |f(x)-L|<\varepsilon\). This is incorrect because it suggests \(\varepsilon\) is chosen before \(\delta\), which is contrary to needing \(\varepsilon\) for every \(\delta\).

Step 3: Analyzing Option B

Option (b) suggests: For every \(\delta>0\), there is a corresponding \(\varepsilon>0\) such that \(0<|x-c|<\varepsilon \Rightarrow |f(x)-L|<\delta\). This is incorrect because it implies the wrong dependency—\(\varepsilon\) depends on \(\delta\) contrary to the proper definition.

Step 4: Analyzing Option C

Option (c) states: For every positive integer \(N\), there is a corresponding positive integer \(M\) such that \(0<|x-c|<1/M \Rightarrow |f(x)-L|<1/N\). This adapts numberical equivalents (\(N\) and \(M\)) but maintains the dependency order and concept of the limit definition, thus it is correct.

Step 5: Analyzing Option D

Option (d) states: For every \(\varepsilon>0\), there is a corresponding \(\delta>0\) such that \(0<|x-c|<\delta\) and \(|f(x)-L|<\varepsilon\) for some \(x\). This is incorrectly phrased because it requires the condition to be true only for 'some' \(x\) rather than 'every', which contradicts the proper definition.

Key Concepts

Epsilon-Delta Definition

The epsilon-delta definition is a formal way to describe the behavior of limits in calculus. This definition is a cornerstone in understanding how functions behave as they approach a specific point. For the limit of a function \(f(x)\) as \(x\) approaches a point \(c\) to be \(L\), we must show that for every number \(\varepsilon > 0\), there is a corresponding number \(\delta > 0\). This means whenever \(0 < |x-c| < \delta\), then it will always follow that \(|f(x) - L| < \varepsilon\).

The procedure works by defining a window around the point \(c\) that gets increasingly smaller. As \(x\) falls within \(\delta\) distance from \(c\), \(f(x)\) should fall within \(\varepsilon\) distance from \(L\). The word "every" in the definition is crucial; it signifies that this relationship holds true no matter how small \(\varepsilon\) is. It intuitively captures the concept of approaching a value without necessarily reaching it.

Calculus

Calculus is the branch of mathematics that studies change. It deals primarily with derivatives and integrals, concepts invented by Isaac Newton and Gottfried Wilhelm Leibniz. Limits are fundamental to calculus as they allow us to define these two key ideas.

• Derivatives measure how a function changes as its input changes. They also determine the slope of the curve at any point.
• Integrals determine the area under a curve, helping to calculate accumulated quantities, like distance traveled over time.

Understanding limits is essential in calculus because they form the foundation of other operations like differentiation and integration. Without a precise and clear concept of limits, calculus theory would be impossible to define and use properly.

Continuity

Continuity is a smoothness feature of functions in calculus. A function is continuous at a point if small changes in the input result in small changes in the output. The function doesn't jump or have breaks at that point.

Formally, a function \(f(x)\) is said to be continuous at a point \(c\) if:\[ \lim_{{x \to c}} f(x) = f(c) \]
This means that as \(x\) approaches \(c\), \(f(x)\) approaches its value at that point, \(f(c)\). Continuity involves limits, and the epsilon-delta definition is often used to prove that a function is continuous at certain points.

• One-sided limits must equal the function value from both directions.
• An interruption, or discontinuity, occurs if \(f(x)\) jumps or has gaps.

Continuous functions are easier to work with, making the transition through calculus operations smooth and predictable.

Mathematical Proofs

Mathematical proofs are logical arguments that verify the truth of a mathematical statement. They are vital in mathematics to ensure accuracy and to confirm that theories, like the epsilon-delta definition of limits, hold true.

In the case of proving a limit using the epsilon-delta definition:

• You start by assuming \(\varepsilon > 0\). Find a \(\delta > 0\) that suits our requirements.
• Show that when \(0 < |x-c| < \delta\), it follows \(|f(x) - L| < \varepsilon\).
• This requires algebraic manipulation and intuitive understanding of the behavior of \(f(x)\).

Proofs are constructed as a chain of logical deductions. They can be detailed and require precision but are ultimately rewarding. They turn abstract concepts into verified statements within mathematics, thereby expanding our understanding and ability to apply such concepts in real-world problems.