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

MATH100 December 2010

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[hide] 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.

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[show] Hint
What are the conditions for a function to be continuous at a point x = a?

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[show] 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.