Question

Find the limit of lim x → (/2) cos(x) / (1 - sin(x)). Use L'Hospital's rule where appropriate. If there is a more elementary method, consider using it.

Answer 1

The limit lim x → (/2) cos(x) / (1 - sin(x)) does not exist.

Step-by-step explanation:

To find the limit of lim x → (/2) cos(x) / (1 - sin(x)), we can use L'Hopital's rule. When we directly substitute x = π/2 into the expression, we get 0/0 form. So, we can differentiate the numerator and denominator separately and find the limit of the resulting expression. Differentiating cos(x) gives -sin(x) and differentiating 1 - sin(x) gives -cos(x). Now, substituting x = π/2 into the differentiated expressions, we get -sin(π/2) / (-cos(π/2)). This simplifies to -1 / 0. Therefore, the limit does not exist.