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^+)(\cos x)/(1-\sin x)`

which yields an indeterminate form
`\frac00`. Rewriting as

`(\cos x(1+\sinx))/((1-\sin x)(1+\sin x))=(\cos x(1+\sin x))/(1-\sin^2x)=(1+\sin x)/(\cos x)`

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