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Why can’t a function graph have two oblique asymptotes as x approaches positive infinity?

Krystian Karczyński

Founder and General Manager of eTrapez.

Graduate of Mathematics at Poznan University of Technology. Mathematics tutor with many years of experience. Creator of the first eTrapez Courses, which have gained immense popularity among students nationwide.

He lives in Szczecin, Poland. He enjoys walks in the woods, beaches and kayaking.


To answer the question posed in the title, we don’t need to go back to the definition of an oblique asymptote, we just need to know what a function is.

As is often the case in mathematics, let’s imagine that the graph of the function HAS two different oblique asymptotes at , and show that assuming this will definitely lead us to a contradiction, so this assumption cannot be accepted.

Graph

In the graph, these asymptotes might look like this:

Graph with two asymptotes And the graph of the function should approach these asymptotes at , so it will look like this:

Graph with two asymptotes and functionAnd what? Can it be like this? Can the graph of a function look like this? Or do we have a problem here?

Problem

Of course, we have a problem. What is shown above cannot be the graph of a function. Let’s go back to the basics, a function by definition is a mapping that assigns exactly one y value to each x argument. And what does our graph show?

Graph with marked x0 argumentWe can see that, for example, the argument is assigned two values – and . And that can’t happen in a function graph, because each x argument must correspond to only one y value.

Therefore, a function cannot have two different oblique asymptotes at . The entire reasoning can be repeated accordingly for 🙂


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