Science:Math Exam Resources/Courses/MATH 180/December 2017/Question 06
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Question 06 

The Buckingham potential where , and are positive constants, is a refinement of the LennardJones potential that describes the potential energy of a diatomic molecule where the atoms are a distance apart. Find and justify all vertical and horizontal asymptotes of .

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

Near the vertical asymptotes, the function tends to (plus or minus) infinity. 
Hint 2 

The horizontal asymptotes are approached by the function for very positive or negative values of . 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Observe that approaches to infinity then either or goes to infinity, as for some . In other words, if is a vertical asymptote of , then is a vertical asymptote either of or of . Recall that the exponential function can never be zero, so is finite for any . On the other hand, the rational function is not defined at , which is the candidate for the vertical asymptote of and hence of . In fact, So is the only vertical asymptote.
For horizontal asymptotes, since is only defined for , we find So the only horizontal asymptote is (if we consider the function as ).
Answer: the vertical asymptote is and the horizontal asymptote is . 