Science:Math Exam Resources/Courses/MATH100/December 2010/Question 04 (a)/Solution 1

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.