trap101 said:So I figured out the solution to lim (x,y)-->(0,y) of Sin(xy)/x, but I figured it out by looking at a solution. I wanted to understand though why with respect to the above limit how Sin(xy)/x = y?
How do you separate the xy in the numerator?
Ray Vickson said:Use l'Hospitals's rule, or else look at the behavior of ##\sin \theta## for small ##|\theta|##.
trap101 said:Using L'Hospital's rule, would I essentially be treating "y" as a constant?
A multivariate limit is a mathematical concept that describes the behavior of a function as it approaches a specific point in a multi-dimensional space. It involves analyzing the changes in multiple variables as they get closer and closer to a given point.
A single-variable limit only involves one independent variable, while a multivariate limit involves multiple independent variables. This means that the behavior of the function can vary in different directions as the variables approach the given point.
A multivariate limit is calculated by evaluating the function at different points that get closer and closer to the given point. This is done for each variable separately, and the resulting values are then analyzed to determine the overall behavior of the function at the given point.
Multivariate limits are used in various fields of science and engineering, such as physics, economics, and computer science. They are particularly useful in analyzing the behavior of complex systems and predicting their future states.
One common misconception is that multivariate limits are always equal to the value of the function at the given point. This is not always true, as the function may have different behavior in different directions. Another misconception is that multivariate limits are only applicable to functions with two or more variables, when in fact they can also be applied to single-variable functions in multi-dimensional spaces.