Question

True or False? Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(\mathbf{F}(x, y)=4 x \mathbf{i}-y^{2} \mathbf{j},\) then \(\|\mathbf{F}(x, y)\| \rightarrow 0\) as \((x, y) \rightarrow(0,0)\)

Short answer

Therefore, the given statement is True. The magnitude \( \|\mathbf{F}(x, y)\| \) of the vector function indeed approaches 0 as the coordinates \( (x, y) \) approach \( (0,0) \).

Step-by-step solution

Identify the Given Vector Function and the Statement

The given vector function is \( \mathbf{F}(x, y) = 4 x \mathbf{i} - y^{2} \mathbf{j} \). The statement we are examining is that the magnitude \( \|\mathbf{F}(x,y)\| \) approaches 0 as \( (x, y) \) approaches \( (0,0) \).

Compute the Magnitude of the Vector Function

The magnitude of a vector function \( \mathbf{F}(x, y) \), denoted \( \|\mathbf{F}(x, y)\| \), is calculated using the formula \(\sqrt{(F_{i})^2 + (F_{j})^2}\), where \( F_{i} \) and \( F_{j} \) are the components of the vector function. So in this case, we get \( \|\mathbf{F}(x, y)\| = \sqrt{(4x)^2 + (-y^2)^2} = \sqrt{16x^2 + y^4} \).

Analyze the Behavior of the Magnitude as (x,y) Approaches (0,0)

To confirm whether the magnitude approaches 0 as \( (x, y) \) approaches \( (0,0) \), we substitute \( x = 0 \) and \( y = 0 \) into the expression for the magnitude. When we do this, we get \( \|\mathbf{F}(0, 0)\| = \sqrt{16(0)^2 + (0)^4} = 0 \). Therefore, as \( (x, y) \) approaches \( (0,0) \), the magnitude indeed tends to 0.