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## Main Question or Discussion Point

So we know that we typically have to use epsilon delta proofs for determining a limit of a multivariable function because there are infinite paths. But can we use removable discontinuities to prove a limit?

Say we want to evaluate the lim( x^2-y^2)/(x+y) as (x,y)->(0,0).

we can factor as follows (x-y)(x+y)/(x+y)

the x+y cancel and we are left with x-y

lim (x-y)=0 as (x,y)->(0,0) since x-y is clearly continuous a continuous function along any path.

Does this work?

What about the path y=-x? Is that entire line a removable discontinuity?

Say we want to evaluate the lim( x^2-y^2)/(x+y) as (x,y)->(0,0).

we can factor as follows (x-y)(x+y)/(x+y)

the x+y cancel and we are left with x-y

lim (x-y)=0 as (x,y)->(0,0) since x-y is clearly continuous a continuous function along any path.

Does this work?

What about the path y=-x? Is that entire line a removable discontinuity?