Intereting Posts

Find the positive $n$ such $2^{n-2i}|\sum_{k=i}^{\lfloor\frac{n-1}{2}\rfloor}\binom{k}{i}\binom{n}{2k+1}$
Choice Problem: choose 5 days in a month, consecutive days are forbidden
Multiplicative property of the GCD
Stolz-Cesaro Theorem, 0/0 Case
Diameter of the Grassmannian
Dimension of set of commutable matrices
Prove Taylor expansion with mean value theorem
Does $\mathsf{ZFC} + \neg\mathrm{Con}(\mathsf{ZFC})$ suffice as a foundations of mathematics?
Why is the absence of zero divisors not sufficient for a field of fractions to exist?
What do I need to know to understand the Riemann hypothesis
Proving the Law of the Unconscious Statistician
Why doesn't using the approximation $\sin x\approx x$ near $0$ work for computing this limit?
Frequency of Math Symbols
Consider the function, f and its second derivative:
Improving my understanding of Cantor's Diagonal Argument

Please help me in computing the following limit.

$$\lim_{(x,y) \to (0,0)}\frac{2x^2y}{x^4 + y^2}.$$

This will be my first attempt in computing a limit involving 2 variables.

Is this a part of multivariable calculus as it contains more than one variable?

How can I interpret this geometrically?

Thank You

- Examples of applying L'Hôpitals rule ( correctly ) leading back to the same state?
- The limit of the alternating series $x - x^2 + x^4 - x^8 + {x^{16}}-\dotsb$ as $x \to 1$
- Prove that $\cos(x)$ doesn't have a limit as $x$ approaches infinity.
- Evaluating $\lim\limits_{x\to \infty}\sqrt{x^{6}+x^{5}}-\sqrt{x^{6}-x^{5}}$
- taking the limit of $f(x)$, questions
- Is $0$ an Infinitesimal?

- Volume of the largest rectangular parallelepiped inscribed in an ellipsoid
- Calculating $\lim_{x\to+\infty}(\sqrt{x^2-3x}-x)$
- How find this limit $\lim_{n\to\infty}\frac{(2n+1)!}{(n!)^2}\int_{0}^{1}(x(1-x))^nf(x)dx$
- parallelizable manifolds
- About $\lim \left(1+\frac {x}{n}\right)^n$
- Proving $ \frac{x^3y^2}{x^4+y^4}$ is continuous.
- Shouldn't this function be discontinuous everywhere?
- Can indefinite double integrals be solved by change of variables technique?
- How to prove that $\int_0^b\Big(\int_0^xf(x,y)\;dy\Big)\;dx=\int_0^b\Big(\int_y^bf(x,y)\;dx\Big)\;dy$?
- Find the limit $\lim \limits_{n\to \infty }\cos \left(\pi\sqrt{n^{2}-n} \right)$

The reason we teach this particular problem is to show that directional limits along straight lines are not enough. Along a straight line $y=mx,$ with $m \neq 0,$ we have

$$ f(x,mx) = \frac{2 x^2 y}{x^4 + y^2} = \frac{2 m x^3 }{x^4 + m^2 x^2} = \frac{2 m x }{x^2 + m^2} $$ from which

$$ |f(x,mx) | = \left| \frac{2 m }{x^2 + m^2} \right| \cdot |x| \leq \left| \frac{2 m }{ m^2} \right| \cdot |x| = \left| \frac{2 }{ m} \right| \cdot |x|. $$

Also, if we take either a vertical line $x=0$ or a horizontal line $y=0$ we get 0.

So, approaching the origin along any **straight line** gives an evident limit of 0. In one variable, that would be enough, but in at least two variables, that is not sufficient to show that there is a limit, just approach the origin along a parabola $y = m x^2$ instead, as in Davide’s answer.

Put $f(x,y):=\frac{2x^2y}{x^4 + y^2}$. Fix a real number $m$. Then for $x\neq 0$

$$f(x,mx^2)=\frac{2x^2mx^2}{x^4+m^2x^4}=\frac{2m}{1+m^2},$$

so if there was a limit, it would be $\frac{2m}{1+m^2}$. These one depends on $m$, which is absurd since the limit would be unique.

As you have pointed out: Putting $(x,y) = (t,t^2)$ will give you:

$$\lim_{t\to 0} \frac{2t^4}{2t^4} = \lim_{t\to 0}1 = 1$$

Using the path $(x,y) = (t,0)$ gives

$$\lim_{t\to 0} \frac{ 2t^2 \cdot 0 }{t^4} = \lim_{t\to 0} 0 = 0$$

Since you got a different value in each case, the original limit cannot exist.

Another approach is to consider the change of variables, $x^2=r \cos \theta$ and $y=r \sin \theta$ then $\frac{2x^2y}{x^4+y^2}=\frac{2r^2 \cos \theta \sin \theta}{r^2}=2\cos \theta \sin \theta$ which depends on $\theta$ and hence the limit doesn’t exist since it’s not unique.

- Conditional probability and the disintegration theorem
- Radical/Prime/Maximal ideals under quotient maps
- Is it possible to formalize all mathematics in terms of ordinals only?
- $\sqrt{17}$ is irrational: the Well-ordering Principle
- Is there any trivial reason for $2$ is irreducible in $\mathbb{Z},\omega=e^{\frac{2\pi i}{23}}$?
- Expressing the integral $\int_{0}^{1}\frac{\mathrm{d}x}{\sqrt{\left(1-x^3\right)\left(1-a^6x^3\right)}}$ in terms of elliptic integrals
- Find $\sum_{i=, j=: i\neq j }\frac{1}{j}$
- Equation with high exponents
- Strong deformation retract
- If $x_n\leq y_n$ then $\lim x_n\leq \lim y_n$
- Motivation behind the definition of tangent vectors
- Convergence of the arithmetic mean
- The series $\sum\limits_{n=1}^\infty \frac n{\frac1{a_1}+\frac1{a_2}+\dotsb+\frac1{a_n}}$ is convergent
- Inequality with exponents $x^x+y^y \ge x^y +y^x$
- Prove that $\sqrt{x}$ is continuous on its domain $[0, \infty).$