Science:Math Exam Resources/Courses/MATH100/December 2010/Question 04 (b)
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Question 04 (b)
Full-Solution Problems. In questions 2-8, justify your answers and show all your work. If a box is provided, write your final answer there. Simplification of answers is not required unless explicitly requested.
NOTE: Another notation for tan-1(x) is arctan(x)
Determine the equations of any asymptotes (horizontal, vertical or slant).
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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
Look up the definitions for vertical and horizontal asymptotes if you are having trouble.
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By inspection of the function, we can see that it is defined everywhere, hence there are no possibilities for vertical asymptotes.
To check if there are any horizontal asymptotes, we must check the behaviour of the function ƒ(x) as x gets very large (both positively and negatively). So we evaluate
Evaluating the limits gives
(Or draw a trigonometric circle to convince yourself of this).
Therefore, there is a horizontal asymptote on the right, given by the line y = 2. It describes the behaviour of the function for large positive x.
Finally, we check for slant asymptotes. Having already found an asymptote on the right (a slant asymptote of slope 0 if you will) we have to check on the left. Since on the left, the function is a polynomial of degree 4, we can directly conclude there is no slant asymptote.
More formally, a function ƒ has a slant asymptote of equation y = mx + h if and only if
(For a polynomial of degree 2 or more, the first limit never converges).
So all in all, there are no vertical asymptotes, no asymptotes (horizontal or slant) on the left and a horizontal asymptote on the right at y = 2.