Problem 82 - Calculus Early Transcendentals

Question

Use the formal definition of a limit to prove that $$\lim _{(x, y) \rightarrow(a, b)} y=b .(\text {Hint}: \text { Take } \delta=\varepsilon$$

Step-by-Step Solution

Answer

Question: Prove that the limit of y as the point (x,y) approaches (a,b) is equal to b using the formal definition of a limit. Answer: We have proved that if 0 < |(x,y)-(a,b)| < ε, then |y - b| < ε by taking δ = ε. Hence, the limit of y as the point (x,y) approaches (a,b) is equal to b, as per the formal definition of a limit: $$\lim _{(x, y) \rightarrow(a, b)} y = b$$.

Step 1: Recall the formal definition of a limit

For any ε > 0, there exists a δ > 0 such that if 0 < |(x,y)-(a,b)| < δ, then |y - b| < ε.

Step 2: Use the hint

We are given a hint to take δ = ε. So, we need to show that if 0 < |(x,y)-(a,b)| < ε, then |y - b| < ε.

Step 3: Analyze the distance

The distance between the points (x, y) and (a, b) is given by the Euclidean distance formula as: $$|(x,y)-(a,b)| = \sqrt{(x - a)^2 + (y - b)^2}$$ Since we have 0 < |(x,y)-(a,b)| < ε, it means that: $$0 < \sqrt{(x-a)^2 +(y-b)^2} < ε$$

Step 4: Use the triangle inequality

We know that the triangle inequality states that for any real numbers x, y, and z: $$|x - y| \leq |x - z| + |z - y|$$ Apply the triangle inequality to our case, where x = y, y = b, and z = a: $$|y - b| \leq |y - a| + |a - b|$$

Step 5: Simplify and compare with ε

We can focus on the term |y - a| + |a - b| and simplify it as: $$|y - a| + |a - b| = |(y-b)+(b-a)| = |(y-b)+(-(a-b))| \leq \sqrt{(x-a)^2 +(y-b)^2}$$ Using the result from Step 3, we know that: $$\sqrt{(x-a)^2 +(y-b)^2} < ε$$ Therefore, we have: $$|(y - b)| \leq \sqrt{(x-a)^2 +(y-b)^2} < ε$$

Step 6: Conclusion

We have shown that if 0 < |(x,y)-(a,b)| < ε, then |y - b| < ε by taking δ = ε. Hence, we have proved that $$\lim _{(x, y) \rightarrow(a, b)} y = b$$ using the formal definition of a limit.

Key Concepts

Formal Definition of a Limit

In calculus, the formal definition of a limit is a foundational concept which introduces precision to the idea of approaching a point. The definition states that for any positive number \(\varepsilon > 0\), there exists a positive number \(\delta > 0\) such that if a point \((x, y)\) is within the distance \(\delta\) of \((a, b)\) without being exactly \((a, b)\), then the value of a function evaluated at \((x, y)\) is within \(\varepsilon\) of its limit value. Mathematically, this is expressed as: if

• \(0 < |(x, y) - (a, b)| < \delta\)

then

• \(|f(x, y) - L| < \varepsilon\)

Understanding this definition is crucial for proving limits, as it provides a structured way to deal with limits in multivariable calculus. It illustrates how we can arbitrarily get close to a limit value \(L\) by making \((x, y)\) suitably close to \((a, b)\).

Triangle Inequality

The triangle inequality is an essential principle in mathematics which states that, for any three points in space, the sum of the lengths of any two sides of a triangle is greater than or equal to the length of the third side. In algebraic terms, for any real numbers \(x\), \(y\), and \(z\), it is expressed as:

• \(|x - y| \leq |x - z| + |z - y|\)

This concept is particularly useful in multivariable calculus when assessing distances and limits. In our particular exercise, the triangle inequality helps in simplifying the expressions obtained from Euclidean distances. By applying it, one can effectively control and bound different parts of our function to stay within specified margin — effectively showing that the function stays within prescribed limits, thus proving our desired limit result.

Euclidean Distance

Within the context of multivariable calculus, the Euclidean distance formula is frequently used to calculate the distance between two points in a plane or space. For two points \((x, y)\) and \((a, b)\), the Euclidean distance is defined as:

• \(\sqrt{(x-a)^2 +(y-b)^2}\)

This formula allows us to determine how far apart the two points are in a straight line. It provides a critical tool for evaluating limits since it connects the notion of "closeness" in multivariable functions. Specifically, in our solution, we used the Euclidean distance to express the condition \(0 < |(x, y) - (a, b)| < \varepsilon\). This helps us determine when we are within the "\(\varepsilon\)-neighborhood" of a point \((a, b)\), a crucial step when applying the formal definition of limits.

Multivariable Limits

Exploring limits in multivariable calculus involves examining the behavior of a function as two or more variables simultaneously approach specific values. Unlike single-variable limits, multivariable limits present additional complexity due to their dependence on more than one variable path toward a point. For function \(f(x, y)\), approaching a point \((a, b)\) implies analyzing all possible paths \((x, y)\) can take to \((a, b)\), and ensuring the function attains a consistent limit value, which remains \(L\). The essence of this verification is straightforward: if every possible path confirms \(f(x, y) \rightarrow L\) as \((x, y) \rightarrow (a, b)\), the multivariable limit holds. Our exercise embraces this complexity by looking at paths defined by Euclidean neighborhoods rather than one-dimensional trajectories alone, offering a robust means to explore and establish limits in a richer, multidimensional setting.