Problem 16 - Calculus Early Transcendentals

Question

The function \(f\) in the figure satisfies \(\lim _{x \rightarrow 4} f(x)=5 .\) For each value of \(\varepsilon,\) find all values of \(\delta>0\) such that $$|f(x)-5|<\varepsilon \quad \text { whenever } \quad 0<|x-4|<\delta\quad(3)$$ a. \(\varepsilon=2\) b. \(\varepsilon=1\) c. For any \(\varepsilon>0,\) make a conjecture about the corresponding value of \(\delta\) satisfying (3)

Step-by-Step Solution

Answer

Answer: The conjecture made about the relationship between ε and δ is that for any ε > 0, the corresponding value of δ is δ = ε.

Step 1: Case (a): \(\varepsilon=2\)

First, let's calculate the value of \(\delta\) when \(\varepsilon=2\). According to the given condition, we want to find the value of \(\delta\) that satisfies: $$|f(x)-5|<2 \quad \text { whenever } \quad 0<|x-4|<\delta$$ By manipulating the inequality, we get: $$-2

Step 2: Case (b): \(\varepsilon=1\)

Now, let's do the same for \(\varepsilon=1\). We want to find the value of \(\delta\) that satisfies: $$|f(x)-5|<1 \quad \text { whenever } \quad 0<|x-4|<\delta$$ By manipulating the inequality, we get: $$-1

Step 3: Case (c): Conjecture

For any \(\varepsilon > 0\), we want to find a \(\delta\) satisfying: $$|f(x)-5|<\varepsilon \quad \text { whenever } \quad 0<|x-4|<\delta$$ Looking at the two cases explored earlier, we see that the value of \(\delta\) is equal to the value of \(\varepsilon\). This suggests that for any \(\varepsilon > 0\), the value of \(\delta\) might be \(\delta = \varepsilon\). Let's make the following conjecture: For any \(\varepsilon > 0\), there exists a \(\delta > 0\) such that: $$|f(x)-5|<\varepsilon \quad \text { whenever } \quad 0<|x-4|<\delta$$ And the corresponding value of \(\delta\) is \(\delta = \varepsilon\).

Key Concepts

Epsilon-Delta Definition

The concept of the epsilon-delta (\(\varepsilon-\delta\)) definition is fundamental in calculus for describing limits. It provides a rigorous way to express the idea that as \(x\) approaches a particular value, the function \(f(x)\) approaches a specific limit. This is crucial in mathematical analysis, and it helps us precisely define what it means for a function to "get close" to a limit value.

This definition states that for every positive number \(\varepsilon\) (epsilon), there exists a \(\delta > 0\) (delta) such that the absolute difference between the function value and the limit is smaller than \(\varepsilon\), whenever the \(x\) values are within a delta-distance from the point, but not equal to the point itself. Mathematically, if \(\lim_{x \to c} f(x) = L\), it means:

• For every \(\varepsilon > 0\), there exists a \(\delta > 0\)
• Such that if \(0 < \left| x - c \right| < \delta\), then \(\left| f(x) - L \right| < \varepsilon\)

This definition ensures that no matter how small we choose \(\varepsilon\), we can always find a \(\delta\) that makes the function values fall within the \(\varepsilon\)-band around the limit. It's like a promise, ensuring that the behavior of the function near a point is predictable and well understood.

Continuity

In calculus, continuity is an important property of a function that describes how the function behaves at and around a point. A function is continuous at a point when there is no interruption or jump at that point; essentially, you can draw the function at that point without lifting your pencil off the paper.

For a function to be continuous at a point \(c\):

• The function \(f(x)\) must be defined at \(c\)
• The limit of \(f(x)\) as \(x\) approaches \(c\) must exist
• The value of the function at \(c\), \(f(c)\), must equal the limit of \(f(x)\) as \(x\) approaches \(c\)

In symbolic terms, \(f(x)\) is continuous at \(c\) if \(\lim_{x \to c} f(x) = f(c)\). This condition ensures a smooth transition as \(x\) moves through \(c\). If a function has these properties at every point in its domain, it is continuous everywhere.

Continuity is crucial because it guarantees that small changes in the input \(x\) result in small changes in the output \(f(x)\). This is key in calculus, as many theorems and applications, such as differentiation and integration, rely on functions being continuous.

Function Behavior Near a Point

The study of a function's behavior near a particular point often involves looking at how values of \(f(x)\) change as \(x\) gets closer to a specified point \(c\). This often involves the investigation of limits.

When analyzing function behavior:

• The limit \(\lim_{x \to c} f(x)=L\) describes how \(f(x)\) behaves as \(x\) approaches \(c\). If this limit exists and equals \(L\), the function approaches a specific value near \(c\)
• This behavior can help us understand and predict how the function will act not only at \(c\) but also near \(c\)

By examining a function's limits, one can determine whether the function is heading towards a peak, a valley, or if it is leveling off. Such analysis is crucial for predicting whether a function is smooth or if it behaves erratically.

In practical terms, understanding function behavior near a point helps in modeling real-world situations where predicting outcomes with minimal error is essential. It also allows engineers and scientists to design systems that tolerate certain fluctuations and maintain operations around specified parameters.

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Key Concepts

Epsilon-Delta Definition

The concept of the epsilon-delta (\(\varepsilon-\delta\)) definition is fundamental in calculus for describing limits. It provides a rigorous way to express the idea that as \(x\) approaches a particular value, the function \(f(x)\) approaches a specific limit. This is crucial in mathematical analysis, and it helps us precisely define what it means for a function to "get close" to a limit value.

This definition states that for every positive number \(\varepsilon\) (epsilon), there exists a \(\delta > 0\) (delta) such that the absolute difference between the function value and the limit is smaller than \(\varepsilon\), whenever the \(x\) values are within a delta-distance from the point, but not equal to the point itself. Mathematically, if \(\lim_{x \to c} f(x) = L\), it means:

• For every \(\varepsilon > 0\), there exists a \(\delta > 0\)
• Such that if \(0 < \left| x - c \right| < \delta\), then \(\left| f(x) - L \right| < \varepsilon\)

This definition ensures that no matter how small we choose \(\varepsilon\), we can always find a \(\delta\) that makes the function values fall within the \(\varepsilon\)-band around the limit. It's like a promise, ensuring that the behavior of the function near a point is predictable and well understood.

Continuity

In calculus, continuity is an important property of a function that describes how the function behaves at and around a point. A function is continuous at a point when there is no interruption or jump at that point; essentially, you can draw the function at that point without lifting your pencil off the paper.

For a function to be continuous at a point \(c\):

• The function \(f(x)\) must be defined at \(c\)
• The limit of \(f(x)\) as \(x\) approaches \(c\) must exist
• The value of the function at \(c\), \(f(c)\), must equal the limit of \(f(x)\) as \(x\) approaches \(c\)

In symbolic terms, \(f(x)\) is continuous at \(c\) if \(\lim_{x \to c} f(x) = f(c)\). This condition ensures a smooth transition as \(x\) moves through \(c\). If a function has these properties at every point in its domain, it is continuous everywhere.

Continuity is crucial because it guarantees that small changes in the input \(x\) result in small changes in the output \(f(x)\). This is key in calculus, as many theorems and applications, such as differentiation and integration, rely on functions being continuous.

Function Behavior Near a Point

The study of a function's behavior near a particular point often involves looking at how values of \(f(x)\) change as \(x\) gets closer to a specified point \(c\). This often involves the investigation of limits.

When analyzing function behavior:

• The limit \(\lim_{x \to c} f(x)=L\) describes how \(f(x)\) behaves as \(x\) approaches \(c\). If this limit exists and equals \(L\), the function approaches a specific value near \(c\)
• This behavior can help us understand and predict how the function will act not only at \(c\) but also near \(c\)

By examining a function's limits, one can determine whether the function is heading towards a peak, a valley, or if it is leveling off. Such analysis is crucial for predicting whether a function is smooth or if it behaves erratically.

In practical terms, understanding function behavior near a point helps in modeling real-world situations where predicting outcomes with minimal error is essential. It also allows engineers and scientists to design systems that tolerate certain fluctuations and maintain operations around specified parameters.