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Chapter 13: Problem 7

Let \(f: \mathbb{R}^{2} \rightarrow \mathbb{R}\) be defined by writing $$ f\left(x_{1}, x_{2}\right)=\frac{x_{1}^{2} x_{2}}{x_{1}^{4}+x_{2}^{2}} \quad f(0,0)=0 $$ Show that \(\lim _{x \rightarrow 0} f(x, m x)=0\) for every \(m \in \mathbb{R}\), but \(f\) is discontinuous at \((0,0)\).

Short Answer

Expert verified
The limit \( \lim_{x \to 0} f(x, mx) = 0 \) for all \( m \), but \( f \) is discontinuous at \((0,0)\).

Step by step solution

01

Substituting the Limit Approach

To analyze the limit as \( x \to 0 \), we substitute \( x_1 = x \) and \( x_2 = mx \). Therefore, we rewrite the function: \[ f(x, mx) = \frac{x^2(mx)}{x^4 + (mx)^2} = \frac{mx^3}{x^4 + m^2x^2}. \]
02

Simplifying the Function

Now, simplify the expression by factoring out \( x^2 \) in the denominator: \[ f(x, mx) = \frac{mx^3}{x^2(x^2 + m^2)} = \frac{mx}{x^2 + m^2}. \]
03

Taking the Limit

Next, compute the limit as \( x \rightarrow 0 \): \[ \lim_{x \to 0} \frac{mx}{x^2 + m^2} = \frac{m \cdot 0}{0^2 + m^2} = 0. \] This shows that the limit is 0 for all real \( m \).
04

Discussing Continuity at \( (0,0) \)

To show discontinuity at \((0,0)\), consider the limit along a different path: evaluate \( f(x, x^2) = \frac{x^4}{x^4 + x^4} = \frac{1}{2} \). The limit is \( \frac{1}{2} \) which is not equal to \( f(0,0) = 0 \). Therefore, \( f \) is discontinuous at \((0,0)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Limits
Limits are fundamental in calculus, especially when dealing with functions of more than one variable. They help us understand the behavior of a function as the input values approach a specific point. When we talk about limits in multivariable calculus, like in our example, we often analyze what happens as each variable approaches a given value simultaneously.
  • Substitution: To evaluate \( \lim_{x \to 0} f(x, mx) \), we notice the path is defined by \( x_1 = x \) and \( x_2 = mx \).
  • Simplification: After substituting, \( f(x, mx) \) becomes \( \frac{mx^3}{x^4 + m^2x^2} \). Factoring helps us see this simplifies to \( \frac{mx}{x^2 + m^2} \).
  • Evaluation: Finally, taking the limit as \( x \to 0 \) shows that \( f(x, mx) \to 0 \) regardless of \( m \).
The key takeaway is that evaluating the limit using a specific path can reveal the function's behavior. This is essential in proving aspects like continuity or discontinuity at specific points.
Multivariable Calculus
Multivariable calculus extends the concepts of single-variable calculus to functions with more than one input. Consider a function \( f: \mathbb{R}^2 \to \mathbb{R} \), which takes two real numbers as inputs and provides a real number as output.
  • Paths: In multivariable calculus, paths represent different ways of approaching a limit. For instance, paths can linearly depend on one variable like \( x_2 = mx \) or take different forms such as \( x_2 = x^2 \).
  • Surface and curves: When evaluating a limit, we envision the function's surface over these paths, and this visualization aids in understanding complex behaviors at certain points.
  • Simplification Techniques: Often, simplifying using algebra before computation is key, as seen with \( \frac{mx}{x^2 + m^2} \).
The interaction of variables and the multitude of paths can make multivariable calculus quite rich. It requires careful consideration of how changes in one variable affect the overall output of the function.
Discontinuity
Discontinuity is a concept that highlights points where a function does not behave smoothly. In multivariable functions, discontinuities can be particularly interesting because they can depend on the paths taken to approach specific points.
  • Path-Dependent Behavior: Our function \( f \) is shown to approach \( 0 \) along linear paths \( (x, mx) \), but changes when you choose a nonlinear path such as \( (x, x^2) \).
  • Mismatch at a Point: Despite continuous limits along specific paths, if any path gives a different result, like \( f(x, x^2) \to \frac{1}{2} \), the point \((0,0)\) becomes discontinuous.
Understanding discontinuity means understanding how a function can vary significantly with small input changes. Even if most paths show one limit, just one exception can establish discontinuity at that point.

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Most popular questions from this chapter

Let \(c_{Q}\) denote the set of all sequences of real numbers converging to a rational number, that is, $$ c_{Q}=\left\\{x \in c: \lim _{i \rightarrow \infty} x_{i} \in \mathbb{Q}\right\\} $$ Is \(c_{Q}\) nowhere dense in \(c ?\) Is \(c_{Q}\) of the first category in \(c ?\)

Prove that a totally bounded metric space must be separable.

Let \(\left\\{f_{k}\right\\}\) be a sequence of real-valued functions on a metric space \(X\). Define what it means for \(\left\\{f_{k}\right\\}\) to approach a function \(f\) uniformly. Show that if each of the functions \(f_{k}\) is continuous and \(f_{k} \rightarrow f\) uniformly on \(X\), then \(f\) is continuous on \(X\).

(The Hilbert Cube) Consider the following two subsets of Hilbert space \(\ell_{2}:\) (a) \(H=\left\\{\left(x_{1}, x_{2}, \ldots\right) \in \ell_{2}:\left|x_{i}\right| \leq 1 / i\right\\}\) (b) \(G=\left\\{\left(x_{1}, x_{2}, \ldots\right) \in \ell_{2}:\left|x_{i}\right|<1 / i\right\\}\) \(H\) is called the Hilbert cube. Show that \(H\) is closed in \(\ell_{2} .\) Is \(G\) open?

In a metric space \((X, d)\), where \(d\) is the discrete metric, determine which sets are nowhere dense, first category, or residual.
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