Science:Math Exam Resources/Courses/MATH110/December 2013/Question 01 (c)

MATH110 December 2013

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Question 01 (c)
Determine whether the following statement is true or false. If it is true, provide justification. If it is false, provide a counterexample. c) $f(x)={\begin{cases}{\frac {3}{x+2}}\,\,&{\text{if}}\,\,x<1\\{\sqrt {x}}\,\,&{\text{if}}\,\,x\geq 1\end{cases}}\,\,\,$ is continuous at all real numbers.

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
You need to check three things: 1. That the function is continue at the break point given by $x=1$ 2. That the right hand side of the function is continuous at all points larger than 1. 3. That the left hand side of the function is continuous at all points smaller than 1.

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. False: To hunt for discontinuities, we have to check inside each piece and also at the boundary. For the boundary to be continuous, we must have: ${\begin{aligned}\lim _{x\rightarrow 1^{-}}f(x)&=\lim _{x\rightarrow 1^{+}}f(x)\\\lim _{x\rightarrow 1^{-}}{\frac {3}{x+2}}&=\lim _{x\rightarrow 1^{+}}{\sqrt {x}}\\{\frac {3}{1+2}}&={\sqrt {1}}\\1&=1.\end{aligned}}$ So the limits exists and is equal to $1$ which happens to be $f(1)=1$. So the function in continuous at the boundary. On the right of $x=1$, the function is continuous since ${\sqrt {x}}$ is only undefined for $x<0$ which is not covered by this case. On the left of $x=1$, the function is discontinuous at $x=-2$ since the denominator is $0$. This fall inside the region considered. So $f(x)$ is discontinuous at $x=-2$. That means $f(x)$ is NOT continuous over all real numbers.

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