The given limit is:

$\lim_{x \to \frac{\pi}{2}} \left( \cos x \right)^{\cos x}$

Step 1: Analyzing the behavior of $\cos x$ near $x = \frac{\pi}{2}$

As $x \to \frac{\pi}{2}$, $\cos x \to 0$.

Thus, the expression becomes:

$\left( \cos x \right)^{\cos x} = 0^{\text{small positive value}}$

While $\cos x \to 0$, $\cos x$ itself remains positive but very small. This results in a form $0^0$, which is an indeterminate form. Hence, we need to apply logarithmic techniques to solve the limit.

Step 2: Taking the logarithm

Define:

$y = \left( \cos x \right)^{\cos x}$

Now take the natural logarithm of both sides:

$\ln y = \cos x \cdot \ln (\cos x)$

We need to evaluate:

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

Step 3: Using limits and approximations

As $x \to \frac{\pi}{2}$, $\cos x \to 0$, so we analyze the limit:

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

We can use the approximation $\cos x \approx \frac{\pi}{2} - x$ for values of $x$ near $\frac{\pi}{2}$. Thus:

$\cos x \ln (\cos x) \to 0 \cdot (-\infty) \text{ (which is of indeterminate form)}$

Step 4: Applying L'Hopital's Rule

This indeterminate form $0 \cdot (-\infty)$ can be converted into a $\frac{0}{0}$ form by rewriting it as:

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

Now, we can apply L'Hopital's Rule by differentiating the numerator and denominator:

• Derivative of $\ln (\cos x)$ is $\frac{-\sin x}{\cos x} = -\tan x$
• Derivative of $\frac{1}{\cos x}$ is $\frac{\sin x}{\cos^2 x}$

So the limit becomes:

$\lim_{x \to \frac{\pi}{2}} \frac{-\tan x}{\frac{\sin x}{\cos^2 x}} = \lim_{x \to \frac{\pi}{2}} -\cos x = 0$

Step 5: Conclusion

Thus, $\ln y \to 0$, which implies that $y \to e^0 = 1$.

Therefore, the original limit is:

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

Final Answer:

$\boxed{1}$

Would you like more details on any specific step?

Here are 5 follow-up questions that expand on this topic:

1. What are the typical forms of indeterminate limits?
2. How does L'Hopital's Rule help in evaluating limits?
3. What other techniques can be used to evaluate limits involving powers like $a^b$?
4. Why is $0^0$ considered indeterminate in limits?
5. Can limits involving trigonometric functions be solved without L'Hopital's Rule?

Tip: Always consider applying logarithmic transformations for complex exponential or power limits, as it simplifies many expressions!