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

Answer:

The limit of the function at x approaches to $\frac{\pi}{2}$ is $-\infty$.

Step-by-step explanation:

Consider the information:

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

If we try to find the value at $\frac{\pi}{2}$ we will obtained a $\frac{0}{0}$ form. this means that L'Hôpital's rule applies.

To apply the rule, take the derivative of the numerator:

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

Now, take the derivative of the denominator:

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

Therefore,

$\lim_{x \to \frac{\pi}{2}}\frac{-sin(x)}{-cos(x)}$

$\lim_{x \to \frac{\pi}{2}}\frac{sin(x)}{cos(x)}$

$\lim_{x \to \frac{\pi}{2}}tan(x)}$

Since, tangent function approaches -∞ as x approaches to $\frac{\pi}{2}$

, therefore, the original expression does the same thing.

Hence, the limit of the function at x approaches to $\frac{\pi}{2}$ is $-\infty$.