Ples me find slant assemtotes this one is different because it isn't a rational function find the slant assemtotes of the equation can be rewritten using the quadratic formula as ples find slant assemtotes and show all work thx

A polynomial asymptote is a function such that





Since this equation defines a hyperbola, we expect the asymptotes to be lines of the form .

Ignore the negative root (we don't need it). If , then we want to find constants such that



We have





since forces us to have . And as , the term is "negligible", so really . We can then treat the limand like



which tells us that we would choose . You might be tempted to think , but that won't be right, and that has to do with how we wrote off the "negligible" term. To find the actual value of , we have to solve for it in the following limit.





We write



Now as , we see this expression approaching , so that



So one asymptote of the hyperbola is the line .

The other asymptote is obtained similarly by examining the limit as .





Reduce the "negligible" term to get



Now we take , and again we're careful to not pick .







This time the limit is , so



which means the other asymptote is the line .

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