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Homework Statement
f(x,y) = (x3+y3)^(1/3)
Show that fy(0,0) = 1
The Attempt at a Solution
fy=y2/(x3+y3)^(2/3)
And...I take the limit of it as x and y goes to zero, which gets me 0/0
Jerbearrrrrr said:If the limit exists (we don't yet know if it does, but the question posed seems to assume it does), then you can evaluate it by choosing a "path" to "lim" along.
The first part of that isn't true- you don't "have to do it along some path". The best way to evaluate limits in two dimensions, as (x, y) goes to (0, 0) is to change to polar coordinates. That way, r alone measures the distance to (0, 0). If the limit, as r goes to 0, does not depend on [itex]\theta[/itex], then the limit exists and is equal to that value.gabbagabbahey said:When you take the limit of a multivariable function, you have to do it along some path [itex]y(x)[/itex]...If every path leads to the same result, then the limit exists and is equal to that result.
A partial derivative limit is a mathematical concept that involves taking the derivative of a multivariable function with respect to one of its variables while holding the other variables constant. It represents the rate of change of the function in a specific direction.
The partial derivative limit is calculated by taking the limit of the difference quotient as the change in the variable approaches zero. This can be done using the limit definition of a derivative or by using specific rules and formulas for partial derivatives.
Partial derivative limits are important in science because they allow us to analyze and understand the behavior of multivariable functions. They are used in fields such as physics, engineering, economics, and more to model and predict the behavior of complex systems.
Yes, a partial derivative limit can exist even if the function is not continuous. This is because the limit only considers the behavior of the function in a specific direction, rather than its overall continuity. However, if the function is not continuous, the limit may not accurately represent the behavior of the function as a whole.
No, partial derivative limits and total derivative limits are not the same. A total derivative limit involves taking the derivative of a function with respect to all of its variables, while a partial derivative limit only considers the derivative with respect to one variable while holding the others constant.