Question

Find the limit. use l'hospital's rule where appropriate. if there is a more elementary method, consider using it. lim x → (π/2)+ cos x 1 − sin x

Answer 1

Looks like the limit is

$\displaystyle\lim_{x\to\pi/2^+}\frac{\cos x}{1-\sin x}$

which yields an indeterminate form $\dfrac00$. Rewriting as

$\dfrac{\cos x(1+\sinx)}{(1-\sin x)(1+\sin x)}=\dfrac{\cos x(1+\sin x)}{1-\sin^2x}=\dfrac{1+\sin x}{\cos x}$

we see the numerator approaches 1 + 1 = 2, while the denominator approaches 0. Since $\cos x$ for $x$ near $\dfrac\pi2$ with $x>\dfrac\pi2$, the limit is $-\infty$.