by-considering-different-paths-of-approach-show-that-the-functions-in-exercises-35-42-have-no-limi-2

Question

By considering different paths of approach, show that the functions in Exercises $35-42$ have no limit as $(x, y) \rightarrow(0,0) .$
$$
f(x, y)=\frac{x^{4}}{x^{4}+y^{2}}
$$
Graph cannot copy

Uma Kumari · Accepted Answer

in this problem were given that function X Y is it was too extra the power forward or extra power forward, less vice. Where now the object is to show that the function has no limits as X Y approaches 00 by considering different parts off approach now, the limit cannot be found by direct substitution because this yields 0/0. So we examined by first part limit X y approaches 00 extra, about four export over actual about four. That&#x27;s why square is equals to limit X approaches. Zero. Expert About four. Our off zero will get limit X approaches zero X to the power forward for the four. We really get one now, considering the second part limit X y approaches 00 x to the power four or X to the power of or less Y square leg moderate X Approaches. Zero at the par four or extra power four plus X square full square. We will get Lim X approaches. Zero extra about four or actually about four doesn&#x27;t about four. No well good limit X approaches zero X to the power four or two extra power for, and they will get 1/2. Therefore, the function has no limits as X Y approaches 00 So that&#x27;s the solution

Uma Kumari · Answer

in this problem, consider the following function. F x y is Cipel&#x27;s toe minus X over under Root X Square plus Vice Square. Now the object is to show that the function has no limit, as ex wife approaches 00 by considering different parts off approach. Now, if the left and right limits exist at the point and are equal then on Lee, the limit off the function at that point exist. Otherwise, it does not exist. Now, along the part that is by is equals, Toe X and X is greater than zero. The limit off the function that is X Y approaches 00 will be limit. X Y approaches 00 minus eggs over under root X square, plus by square, the equal toe limit X approaches. Zero positive. And why is it was toe X minus X or X square plus X square. It should be equal toe limit. X approaches zero positive minus X Our under two X square equals toe limit. X approaches zero positive minus X over. I know two more eggs now Use more X is equals to eggs. You X is greater than zero than we will get. Limit X approaches. Zero positive minus X or under two X, which will be equal toe limit X approaches zero positive minus one over under two and minus one or under. Rule two. Now along the path that will be by physicals. Toe eggs on X is less than zero. We will get limit X Y approaches 00 minus eggs or under rude X squared plus y square is equals toe limit X approaches. Zero negative and why is equals to eggs minus X over. Under Ruth, X square plus X square is equals toe limit X approaches zero negative minus X over under two X square is equals toe limit X approaches zero negative minus eggs. Our under two more eggs now use more exes equals toe eggs. If X is less than zero, then we will get Lim X approaches the O negative minus eggs under two minus eggs. We will get limit. X approaches the O negative one hour Who to which will be one hour route to now use the two results. It confirms that both results are defend, so the limit off the function does not exist at X y approaches 00 Therefore, the function has no limit at X y approaches 00 So that&#x27;s a solution

Uma Kumari · Answer

in this problem, consider the function F X y physicals, toe extra, about four minus y square or actually the power four plus twice where. How the objective is to show that the function has no limit by considering different parts of approach on the limit cannot be found by direct substitution, which gives the indeterminant form that will be 0/0. Now examine the limit off the function along the different parts. That is why is equals to K X square so that so take the limit as limit X Y approaches 00 alone. Why is equals toe K X Square, which gives us f x y Z equals toe limit. X y approaches 00 along. Why is it was to be expire X to the power four minus vice square over. It&#x27;s about four. Bless my square now, putting the values or pie, we will get limit X approaches zero X to the power four minus K X square, Whole square over. Excited about four plus a X Square whole square now, But simply find this. We will get limit. X approaches zero X to the path forward one minus games. Where over extra about four bunless is where we will get one minus case where over one plus is where therefore the value off the limit depends on the value off constant K. Hence the value off limit is wearing with the value off. Hey, now for K is equals to one the value X fight approaches 00 along the parabola. Why is it was two x square and the limit is limit. X y approaches 00 along Why is equals two x square f x y is It was toe one minus one whole square over one plus one whole square because her key is it was 21 By solving this really get zero over to that is it was 20 Hence for K is equals to one that along the parabola why is equal to X square and the limit is zero. Now we&#x27;ll ask poor he is equals to zero the value exp I approaches 00 along the x X is Why is it was 20 and the limit will be limit. X y approaches 00 along why is equals to zero f x y is equals toe one minor, zero whole square over one plus zero whole square. Because here K is equals to zero by solving the civilian one or one is equals to one hand, or K is equals to zero. Along the X axis is why is equals to zero and the limit is one. Therefore, the function F x y have different limits or different values. Uh, yeah, hands the limit off function. F X Y does not exist, so that&#x27;s the solution.

Ahmed Ibrahim · Answer

the solution off this question will start by will divide the solution to reports first part along Why equals you don&#x27;t. So we&#x27;ve been sub secure it. Why equals zero in dysfunction So we will have exposed for over export for close to you So a solution equals one also weak The other part as Lim x And why approaches zero along Why equal excess square so way have Sorry we have export for over X or for us Why post too? So after substituting why Collection square We will have explored for over exports four plus export for so the solution will be won over to thank you.

Uma Kumari · Answer

in this problem, consider the fallen function. F X Y is equals two X y or model X Y off. The objective is to show that about function has with no limit as X Y approaches 00 By considering different parts of approach, the limit cannot be found by direct substitution. Because this is 0/0 along the girl. Why is it was okay. X X is not equal to zero before the limit becomes X Y approaches 00 alone. Why is equals K X and he is not ableto zero x y Oh, a model ex wife is it was toe limit Eggs approaches zero ex military icky eggs. Oh, uh, model X multiply GX, which will give limit X approaches zero k x square model K X square, which will give us limit X approaches zero j x square over model. Okay, Malita X square and finally gives us Lim X approaches. Zero Okay, over more. Okay, So if the value off K is greater than zero, then the limit will be one. And if that day is less than zero, then the limit will be minus one. That is, the limits are different than the approach is different. Therefore, the function f x y is equals toe X Y or Model X Y has no limits as X Y approaches 00 so that&#x27;s a solution.

Uma Kumari · Answer

consider the following function h of X comma by is equals toe Take square by a Paul X to the power four plus by square. The objective is to show that the above function has no limits as X comma vitamins to zero comma zero by considering different part off approach. The limit cannot be found by direct substitution method because by direct substitution, which gives the 00 indeterminate for now, examine the value of F along parabolic alot that ain&#x27;t at zero comma. Zero along the curve eyes equals two key X were coma access not equals to zero. The above function becomes h of X comma by this equals toe X square into why divided by X to the power of whole plus Vice square, which is equals two X where in two cakes where have divided blood x to the power full plus k x squared whole square By simply find it, we get que upon one plus k square, then lim X comma vitamins to zero comma zero along buys equals to play it square. It affects comma by is equal to lim X comma vitamins to zero comma zero along bicycles to take square que upon one plus case where it is equals toe que upon one plus K square. The limit varies with the part off approach. If X comma via tends to zero comma zero along the parable Love I is equal to X square that is que is equals to one. Then the limit is when upon one plus one square is equals to one upon to If x comma violence to zero comma zero along the X axis, that is why is equals to zero. It means that case also equals to zero than the limit has zero upon one plus zero, which is equals to zero by the two path test. If has no limit as it&#x27;s common guidance to zero comma zero does the given function has no limit as X comma vitamins to zero comma zero

Problem

By considering different paths of approach, show …

Problem 36 Medium Difficulty

By considering different paths of approach, show that the functions in Exercises $35-42$ have no limit as $(x, y) \rightarrow(0,0) .$
$$
f(x, y)=\frac{x^{4}}{x^{4}+y^{2}}
$$
Graph cannot copy

Answer

the function has no limit as $(x, y) \rightarrow(0,0)$

Related Courses

Thomas Calculus

Related Topics

Discussion

Top Calculus 3 Educators

Caleb E.

Kristen K.

Michael J.

Joseph L.

Calculus 3 Courses

Vectors Intro

Vector Basics - Example 1

Recommended Videos

Watch More Solved Questions in Chapter 14

Video Transcript

George B. Thomas, Maurice D. Weir, Joel Hass, Frank R. Giordano

Thomas Calculus

Caleb E.

Kristen K.

Michael J.

Joseph L.

Vectors Intro

Vector Basics - Example 1

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

the function has no limit as $(x, y) \rightarrow(0,0)$

Thomas Calculus

Caleb E.

Kristen K.

Michael J.

Joseph L.

Vectors Intro

Vector Basics - Example 1

George B. Thomas, Maurice D. Weir, Joel Hass, Frank R. GiordanoThomas Calculus

Caleb E.

Kristen K.

Michael J.

Joseph L.

George B. Thomas, Maurice D. Weir, Joel Hass, Frank R. Giordano

Thomas Calculus