lim x → π/2 (1 - sin x) / (cos x)

Steps to Solve

1. Substituting the limit directly

We begin by substituting $x = \frac{\pi}{2}$ directly into the expression:

$$ \frac{1 - \sin\left(\frac{\pi}{2}\right)}{\cos\left(\frac{\pi}{2}\right)} = \frac{1 - 1}{0} = \frac{0}{0} $$

This results in an indeterminate form ( \frac{0}{0} ).

2. Applying L'Hôpital's Rule

Since we have an indeterminate form, we can use L'Hôpital's Rule, which states that if the limit evaluates to ( \frac{0}{0} ) or ( \frac{\infty}{\infty} ), we can take the derivative of the numerator and the derivative of the denominator:

The derivative of the numerator $(1 - \sin(x))$ is:

$$ \frac{d}{dx}(1 - \sin(x)) = -\cos(x) $$

The derivative of the denominator $(\cos(x))$ is:

$$ \frac{d}{dx}(\cos(x)) = -\sin(x) $$

3. Re-evaluating the limit

Now we can evaluate the limit again using these derivatives:

$$ \lim_{x \to \frac{\pi}{2}} \frac{-\cos(x)}{-\sin(x)} = \lim_{x \to \frac{\pi}{2}} \frac{\cos(x)}{\sin(x)} = \frac{\cos\left(\frac{\pi}{2}\right)}{\sin\left(\frac{\pi}{2}\right)} = \frac{0}{1} = 0 $$