Now, I will slightly change the function from the previous example. This time let f(x,y) be given by the formula, when x times y is divided by the square root of x squared plus y squared. Once again, since we're not allowed to divide by 0, we exclude the origin. The question is as before whether this particular function is the limit when xy approach the origin. Let me remind you that from Cartesian coordinates x and y, we can change them into polar coordinates by the formulas. Let me draw a coordinate plane. Suppose we have the point here, whose coordinates are x and y. Now, if we join this point with the origin by a segment of a straight line or we can think this is a vector, then this is an angle which we denote it by Phi letter, Greek letter Phi. Now, r is the length of this vector. So, r equals, from the right triangle, the sum of the sides plus the length of the hypotenuse and Phi is the angle drawn here. Quite often, the change of the variables from Cartesian to polar are quite useful and helpful in finding the limit. Now when we replace x and y by the polar coordinates, substituting into the arguments on the function f, we have f, we get, after simplifying what's left. Now, what it means that x and y tend to origin, x y tend to origin (0,0) and here we have f, that means that r also tends to 0, Now, here we have the function, I can repeat here r tends to 0 and we have r times sine Phi cos Phi. Now, recall from simple calculus that there is a Sandwich Theorem for the sequences, this theorem claims that given three sequences which are in the double inequality, sequences of real numbers xn, yn, and zn. Let's suppose the very left sequence tends to a and the very right tends to a, now from this theorem it follows that yn, which is in between that's why is called Sandwich Theorem, also should tend to a. We can use it here although we have no sequences here. Still, we have a function which is a product of the radius r which is becoming progressively smaller and smaller, and we have a product of two trigonometric functions each of them is a bounded function. We now Sine Phi as well as cosine Phi the values of these functions never exceed positive one and never are less than negative one and positive one, so they lay in between. That means that if I write the absolute value taking into account the boundedness of these functions I have simply are here, and the absolute value is always non-negative and this tends to 0. So, using the Sandwich Theorem or rather this analog, we can say that the limit exists and it equals 0. So, this function tends to 0. Now, after this introduction about limits, let's define continuity and we'll define this property of a function at a given point. So, let's suppose we have a function of n variables and D is domain, and a particular point x0 belongs to domain. Function f(x) is continuous at this point if the limit of this function exists as x approaches 0 and it equals f of (x0). If function f(x) is continuous at any point of D, then we call it continuous on D. Continuous functions play a very important role in economic analysis and this is the right time to recall the Weierstrass Theorem. Remember I said that if a function is given and this function is a function of one variable and is continuous on a closed interval, so we're recalling the theorem from single variable calculus, hearing by Weierstrass. Let f(x) be continuous on [a,b] closed interval, then they exist numbers and such that the function is bounded, it's values lies in between lowercase m and capital M and these values are attainable because f(x1) is m and f(x2) is capital M. The same theorem or rather similar theorem holds for n-dimensional case but this time we have no segments in the n-dimensional space, so we instead have compact sets. So, if it's a statement you replace the domain which is a segment by the compact set in rn, then we can keep the rest. So, this theorem claims that there will be two values, lowercase m, capital M such that the value of the function f lies between these two values and there exist two points from the domain x1x2. This time we have to write instead of subscripts, upscript such that the function at one point takes the least value and in another point it takes the greatest value.