# Thread: limit as h, k approach 0 of h^2 + hk + k^2

1. ## limit as h, k approach 0 of h^2 + hk + k^2

i am trying to determine what is $\lim_{\substack{h\rightarrow 0\\k\rightarrow 0}} h^2 + hk + k^2$

at first i thought the limit was just 0 but i am trying to be careful now to make sure. the only methods i know to show that a multivariable limit exists is to apply the squeeze theorem. however in this case i can't seem to find any way to do it. i then suspect that this limit doesn't exist but likewise i can't seem to find 2 curves which give different limits. it seems that any polynomial i travel on will make the limit 0. i also tried reciprocal functions like h = 1/k but those don't work either since if k approaches 0 then h obviously does not. i am not sure what else i can try. any ideas to show that the limit exists or doesn't exist?

2. ## Re: limit as h, k approach 0 of h^2 + hk + k^2

Originally Posted by oblixps
i am trying to determine what is $\lim_{\substack{h\rightarrow 0\\k\rightarrow 0}} h^2 + hk + k^2$

at first i thought the limit was just 0 but i am trying to be careful now to make sure. the only methods i know to show that a multivariable limit exists is to apply the squeeze theorem. however in this case i can't seem to find any way to do it. i then suspect that this limit doesn't exist but likewise i can't seem to find 2 curves which give different limits. it seems that any polynomial i travel on will make the limit 0. i also tried reciprocal functions like h = 1/k but those don't work either since if k approaches 0 then h obviously does not. i am not sure what else i can try. any ideas to show that the limit exists or doesn't exist?
All polynomials are continuous, so you can just substitute those two values.