Science:Math Exam Resources/Courses/MATH105/April 2011/Question 09 (a)

MATH105 April 2011

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Question 09 (a)
Each of the short-answer questions below is worth 5 points. Put your answer in the box provided and show your work. No credit will be given for the answer without the correct accompanying work. Does $\lim _{(x,y)\rightarrow (0,0)}{\frac {xy^{2}}{x^{2}+y^{4}}}$ exist? If it does, find its value. If not, write "does not exist" in the box and give reasons why.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
Recall that if a limit exists then it has a unique value. Therefore, if you can show that the limit takes on at least two different values then it cannot exist.

Hint 2
One can quickly see that approaching the origin from an axis, when x or y is zero, then the expression is zero. Hence, if the limit exists, then it must be zero.

Hint 3
Can you think of a path towards the origin such that the limit is non-zero? One way to achieve this would be to choose x and y such that the enumerator and denominator have the same power.

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. No, this limit does not exist. We can see this by considering two different paths to the origin when taking this limit. Let $f(x,y)={\frac {xy^{2}}{x^{2}+y^{4}}}.$ First, consider the given limit when the limit is taken by approaching on the $x$-axis. Notice in this case we have f(x,0)=0 for all x. Therefore, $\lim _{(x,y)\rightarrow (0,0)}f(x,y)=\lim _{x\rightarrow 0}f(x,0)=\lim _{x\rightarrow 0}0=0.$ Now consider a different path. In order to find a path that makes the limit non-zero, we choose x and y such that the powers of the enumerator and denominator are equal. The highest power in the denominator is the 4 in y. So let us choose x such that the enumerator matches that power: Let us approach the origin along the curve $x=y^{2}$. $\lim _{(x,y)\rightarrow (0,0)}f(x,y)=\lim _{y\rightarrow 0}f(y^{2},y)=\lim _{y\rightarrow 0}{\frac {y^{4}}{y^{4}+y^{4}}}={\frac {1}{2}}\neq 0.$ So, depending on the path taken to the origin, we get a different value for the above limit. Therefore, the limit does not exist.

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