00:01
Hi there, in this question, these are the limit questions.
00:04
So let's start with part a.
00:07
So there are two variables of x, y.
00:10
So this is equal to xy divided by x minus 1 and times y minus 1.
00:21
So we have to look, there is in such a limit at x is equal to 0 .0.
00:27
That means this is the origin.
00:28
So we have to just find the value when x and x goes to n y goes to zero.
00:38
Let's do direct substitution here, which is equal to on the top we have 0 times 0 divided by 0 minus 1, negative 1 times 0 plus 1, which is 1, that is equal to 0 over negative 1, which is equal to 0.
00:52
So we got a real number.
00:54
That means the limit value for this function is equal to 0.
00:59
And in part b, there is a piecewise function.
01:04
So we have to, for the limit of the function at the origin, we have to use this rule here, x, y, fourth, and x to the fourth plus y to the fourth.
01:17
So first of all, we have to do the direct substitution here.
01:21
When x goes to zero and y goes to zero, let's take a look at what value we got.
01:26
This is 0 times 0 to 4th, which is 0, and 0 to 4th, 0.
01:30
0 plus 0, which is 0.
01:31
So this is an indeterminate form.
01:34
So what we have to do in this case.
01:36
So we have to just take a look at coordinate plane.
01:43
So we are approaching the origin here.
01:47
First of all, let's approach along the x -axis.
01:50
So along the x -axis, the y -value value is 0.
01:53
So i can just say along the x -axis, the y -value is equal to 0.
01:58
So that means when limit x -goes goes to 0.
02:02
So i'm going to just plug in 0 for y value, which is x times 0 to the 4th divided by x to the 4th plus 0 to 4th.
02:11
And we can just get that x goes to 0, which is 0 over x to the 4th.
02:19
If you divide 0 over x to the 4th, we will get 0.
02:22
And we have to look at along the y -axis here.
02:26
So along the y -axis, that means the x values are, so we are approaching from the y -end.
02:33
Axis here.
02:34
So along the y -axis, the x values are zero.
02:37
That means axis equal to zero.
02:39
So we have to just limit y goes to zero.
02:43
So we have to plug in zero in this case.
02:45
This is zero times y to the fourth, zero to the fourth plus y to the fourth.
02:51
And we got limit y goes to zero, which is y to the fourth divided by not five to the fourth, zero times y to the fourth, which is zero...