Show that the limit as (x, y) approaches (0, 0) of (2xy)/(x^2 + y^2) does not exist.


Show that {eq}\displaystyle \; \lim_{(x, y) \rightarrow (0, 0)} \frac{2xy}{x^2 + y^2} \; {/eq} does not exist.

Limits Along Two Paths:

To prove that the limit {eq}\displaystyle\lim_{(x,y) \rightarrow (a,b) } f(x,y) {/eq} does not exist, we must think of two different paths through the point {eq}(a,b) {/eq} along which the limit produces two unequal results.

For the given limit, we will employ the paths {eq}y=0 {/eq} and {eq}y=x {/eq}.

Answer and Explanation:

We calculate the value of the given limit along the paths {eq}y=0 {/eq} and {eq}y=x {/eq}.

To determine the limit along the path {eq}y=0 {/eq}, we...

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Learn more about this topic:

How to Determine if a Limit Does Not Exist

from AP Calculus AB: Exam Prep

Chapter 4 / Lesson 9

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