The expression in the image is a limit problem, specifically:

$\lim_{\theta \to \frac{\pi}{4}} \frac{\sin(\theta) - \frac{\sqrt{2}}{2}}{\theta - \frac{\pi}{4}}$

To solve this, we observe that direct substitution of $\theta = \frac{\pi}{4}$ into the expression will lead to the indeterminate form $\frac{0}{0}$. This suggests that we can apply L'Hôpital's Rule, which helps solve limits that result in indeterminate forms.

Steps to solve:

1. Differentiate the numerator: The numerator is $\sin(\theta) - \frac{\sqrt{2}}{2}$. The derivative of $\sin(\theta)$ is $\cos(\theta)$.

2. Differentiate the denominator: The denominator is $\theta - \frac{\pi}{4}$. The derivative of $\theta$ is simply 1.

Now, apply L'Hôpital's Rule:

$\lim_{\theta \to \frac{\pi}{4}} \frac{\sin(\theta) - \frac{\sqrt{2}}{2}}{\theta - \frac{\pi}{4}} = \lim_{\theta \to \frac{\pi}{4}} \frac{\cos(\theta)}{1}$

Substituting $\theta = \frac{\pi}{4}$:

$\frac{\cos\left(\frac{\pi}{4}\right)}{1} = \frac{\frac{\sqrt{2}}{2}}{1} = \frac{\sqrt{2}}{2}$

Final Answer:

$\lim_{\theta \to \frac{\pi}{4}} \frac{\sin(\theta) - \frac{\sqrt{2}}{2}}{\theta - \frac{\pi}{4}} = \frac{\sqrt{2}}{2}$

Would you like more details about how this rule works, or do you have other questions related to this topic?

Here are 5 related questions:

1. What is the general form of L'Hôpital's Rule?
2. Can limits like this one be evaluated using other methods besides L'Hôpital's Rule?
3. How does $\cos(\theta)$ behave near $\frac{\pi}{4}$?
4. What is the geometric interpretation of this limit in terms of slopes?
5. What happens if we apply this limit for other values of $\theta$?

Tip: L'Hôpital's Rule is a powerful tool for resolving limits that yield indeterminate forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$.