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

MATH100 December 2010

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Question 04 (a)

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) Show that ƒ(x) is continuous at x = 1.

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
What are the conditions for a function to be continuous at a point x = a?

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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. To have continuity we need that: [1] The function f(x) is defined at x = 1. [2] $\lim _{x\rightarrow 1}f(x)$ exists and is equal to f(1). Clearly we can evaluate $f(1)={\frac {4}{\pi }}\tan ^{-1}(1)=1$. So f(1) is defined and [1] is satisfied. To check condition 2), we need to evaluate the limit as x goes to 1 of f(x), but since the function is piecewise defined, we need to evaluate the left and right-hand limits and confirm that they are equal. ${\begin{aligned}\lim _{x\to 1^{-}}f(x)&=\lim _{x\to 1}\left({\frac {4}{\pi }}\tan ^{-1}(x)\right)={\frac {4}{\pi }}\tan ^{-1}(1)=1\\\lim _{x\to 1^{+}}f(x)&=\lim _{x\to 1}\left(2-x^{4}\right)=2-(1)^{4}=1\end{aligned}}$ From this we can see the left and right-hand limits are equal and that they equal f(1), so [2] is also satisfied. Thus f(x) is continuous at x = 1.

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