Science:Math Exam Resources/Courses/MATH100/December 2010/Question 04 (b)

MATH100 December 2010

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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. Let $f(x)={\begin{cases}{\frac {4}{\pi }}\tan ^{-1}x&{\text{if }}x\geq 1\\2-x^{4}&{\text{if }}x<1\end{cases}}$ 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!

Hint
Look up the definitions for vertical and horizontal asymptotes if you are having trouble.

Checking a solution serves two purposes: helping you if, after having used the hint, 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. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. 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 $\lim _{x\to +\infty }f(x),\quad \lim _{x\to -\infty }f(x).$ Evaluating the limits gives ${\begin{aligned}\lim _{x\to -\infty }f(x)&=\lim _{x\to -\infty }2-x^{4}=-\infty \\\lim _{x\to \infty }f(x)&=\lim _{x\to \infty }{\frac {4}{\pi }}\tan ^{-1}(x)={\frac {4}{\pi }}\cdot {\frac {\pi }{2}}=2\end{aligned}}$ Note: that $\lim _{x\to +\infty }\tan ^{-1}(x)={\frac {\pi }{2}}$ since $\lim _{x\to \pi /2}\tan(x)=+\infty$ (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 $\lim _{x\to \infty }{\frac {f(x)}{x}}=m\qquad \lim _{x\to \infty }f(x)-mx=h$ (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.

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