Finding limit of composition functions

Question

Consider $\lim_{x \to 2^{+}}f(x) = 3$ and $\lim_{x \to 2^{-}}f(x) = 1$. Now find $\lim_{x \to \frac{\pi}{4}} f(\tan x + \cot x)$.

My try : I tried using the limit rule for composite functions but it isn't continuous at $x=2$ .

Answer 1

As mentioned in the comments, $\tan(x)+\cot(x)$ has a minimum near $\pi/4$. You could show it as follows - first find stationary points of the expression: $$\frac{d}{dx}(\tan(x)+\cot(x))=\sec^2(x)-\csc^2(x).$$ This is $0$ when $\sec^2(x)=\csc^2(x)$. So $\tan^2(x)=1$. In the range $x\in(0,\pi/2)$, this is only true at $x=\pi/4$. You can then show that this is a minimum by for instance considering the values either side, so $$\tan(x)+\cot(x)\ge\tan(\pi/4)+\cot(\pi/4)=2$$ in the range $x\in(0,\pi/2)$.

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So in the limit as $x\rightarrow\pi/4$, $\tan(x)+\cot(x)\rightarrow2^{+}$ so the limit you require is 3.