## University Calculus: Early Transcendentals (3rd Edition)

$$\lim_{x\to(\pi/2)^-}\tan x=\infty$$
$$A=\lim_{x\to(\pi/2)^-}\tan x=\lim_{x\to(\pi/2)^-}\frac{\sin x}{\cos x}$$ As $x\to(\pi/2)^-$, $\sin x$ approaches $\sin(\pi/2)=1\gt0$, while $\cos x$ approaches $\cos(\pi/2)=0$ from the left, where $\cos x\gt0$. (The left side of $\pi/2$ are the values of $x=\pi/3, \pi/4$, etc. And as we consider only values close $\pi/2$, all these values of $x$ are positive) Since as $x\to(\pi/2)^-$, $\sin x\gt0$ and $\cos x\gt0$, so $\sin x/\cos x$ will approach $\infty$. Therefore, $$A=\infty$$