Question

!! AP CALCULUS AB !! find the limit of [(secx*sinx) + (cscx*cosx)]/(3secx) as x approaches 3pi/2

Answer 1

$f(x)= \frac{\frac{\sin{x}}{\cos{x}}+\frac{\cos{x}}{\sin{x}}}{3\sec{x}}$

$\frac{\frac{\sin{x}}{\cos{x}}+\frac{\cos{x}}{\sin{x}}}{3\sec{x}} \implies \\ \frac{\sin^2{x}+\cos^2{x}}{\sin{x}\cos{x}}\frac{\cos{x}}{3}\implies \\\frac{1}{\sin{x}\cos{x}}*\frac{\cos{x}}{3}\implies \\ \frac{1}{3\sin{x}}$

So,

$\lim_{x \rightarrow \frac{3\pi}{2}}{f(x)=f(\frac{3\pi}{2})=\frac{1}{3\sin{\frac{3\pi}{2}}}=-\frac{1}{3}$