limit (x^2+y^2)/(x+y)^2 as (x,y)->(0,0)
I don't know how to prove it doesn't exist. Please help!
Consider the line through the points (3,2,5) and (1,1,1). Consider the plane A that is perpendicular to this line, and passing through the point (-1,0,2). Consider the plane B that passes through the points (1,-1,0), (0,2,0), and (0,5,2). Consider the intersection of plane A and plane B. Write the equation of the line of intersection.
type your equation onto the website wolfram alpha ... google the website if you're unfamiliar with it.
along x=y, the expression is
2x^2/4x^2 = 1/2
along x=2y, we have
5y^2/9y^2 = 5/9
In other words, as we approach (0,0) along different paths, the limit varies, so cannot be said to exist as a single number.
To determine whether the limit of a function exists as (x, y) approaches a specific point, you can try approaching it along different paths and see if the limit is consistent. If the limit is the same along all paths, then it exists.
In this case, to evaluate the limit as (x, y) approaches (0, 0) of `(x^2 + y^2) / (x+y)^2`, consider approaching the point along the x-axis, the y-axis, and the line y = mx where m is any real number.
Approaching along the x-axis: Set y = 0. The limit simplifies to `(x^2 + 0) / (x+0)^2 = x^2 / x^2 = 1`.
Approaching along the y-axis: Set x = 0. The limit simplifies to `(0 + y^2) / (0+y)^2 = y^2 / y^2 = 1`.
Approaching along the line y = mx: Substitute y = mx into the original expression. The limit becomes `(x^2 + (mx)^2) / (x + mx)^2`. Simplifying, we have `(x^2 + m^2x^2) / (x + mx)^2`. Factoring out an x^2, `(1 + m^2) / (1 + m)^2`.
Now, let's consider what happens when x approaches 0. Regardless of the value of m, the numerator remains constant, but the denominator approaches 0. This means that the limit as x approaches 0 of `(1 + m^2) / (1 + m)^2` depends on the value of m.
Since the limit is not the same along all paths approaching (0, 0), the limit of `(x^2 + y^2) / (x+y)^2` does not exist as (x, y) approaches (0, 0).