Science:Math Exam Resources/Courses/MATH110/December 2010/Question 01 (a)

MATH110 December 2010

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[hide] Question 01 (a)
Determine whether each statement is true. If it is, explain why; if not, give a counter-example. Given polynomials ƒ(x) and g(x), $\lim _{x\to \infty }{\frac {f(x)}{g(x)}}$ is equal to 0, $\infty$ or $-\infty$ .

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.

[show] Hint 1
If you believe the statement is true, you need to show that for any two polynomials the limit is always either 0 or $\displaystyle \pm \infty$ . If you believe the statement is false, you must give one example of two polynomials f and g where the limit is neither 0 nor $\displaystyle \pm \infty$ . When trying these types of problems, it is always a good idea to try a few examples out first before trying to solve the problem just to get a feeling of what is required.

[show] Hint 2
Remember that the infinite limit of a function gives its horizontal asymptotes. Is it possible for a function $f(x)/g(x)$ to have a horizontal asymptote NOT at $0$ ?

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.

If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work.
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[show] Solution
The statement is false. Let's take polynomials that not only have a nonzero horizontal asymptote but that also always obtain their asymptote! We can choose constant functions for f and g (these are polynomials!) So suppose that $\displaystyle f(x)=g(x)=1$ . Then, the limit becomes $\displaystyle \lim _{x\to \infty }{\frac {f(x)}{g(x)}}=\lim _{x\to \infty }{\frac {1}{1}}=1$ which is not 0 and does not diverge to $\pm \infty$ . Note: If, for some reason, you don't like constant polynomials there is a whole zoo of non-constant polynomials f and g with quotients approaching any horizontal asymptote you want, e.g. $\displaystyle \lim _{x\to \infty }{\frac {17x+1}{x}}=17.$