Science:Math Exam Resources/Courses/MATH110/April 2013/Question 02 (b)

MATH110 April 2013

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Question 02 (b)
Sketch a single function satisfying all of the listed criteria. $\lim _{x\rightarrow -1^{-}}f(x)=0$ $\lim _{x\rightarrow -1^{+}}f(x)=\infty$ $\lim _{x\rightarrow 1}f(x)=-\infty$

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.

Hint 1
What does it mean when a limit evaluates to a number? To infinity? Each limit listed above turns into a graph feature like a point or asymptote.

Hint 2
Each limit translates into the following graph feature: Approaching x = -1 from the left, the graph approaches the point (-1,0). There is a vertical asymptote at x = -1 and the function increases to $\infty$ as it approaches x = -1 from the right. There is a vertical asymptote at x = 1 and the function decreases to $-\infty$ as it approaches x = 1 from either side.

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. 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. See the second hint for how each limit translates into a feature of the graph. Your graph can do whatever it wants to the left of x = -1, so long as it approaches the point (-1,0). The simplest (yet somewhat boring) way to achieve this is by setting the function to 0 for x < -1. Between x = -1 and x = 1 the function has to fall from $\infty$ to $-\infty$. To the right of x = 1 the function has to come back from $-\infty$ and then can do whatever you want it to do. There is a lot of freedom in how your graph may look, but as long as you follow the key points above your graph will be correct. Below is our example (the purple lines represent asymptotes):

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