91Ó°ÊÓ

To fully grasp the given exercise, understanding trigonometric identities is key.
One important identity is \(\tcos^2 x + \tsin^2 x = 1\).
This identity tells us that if we know the value of \(\tcos x\), we can easily find \(\tsin x\), and vice versa.
For the given problem, we use this identity to simplify the numerator.
By rewriting \(\tcos^2 x\) using the identity, we have \(\tsin^2 x = 1 - \tcos^2 x\),
thus transforming \(\frac{1-\tcos^2 x}{2 x^{2}}\) into \(\frac{\tsin^2 x}{2x^2}\).
This crucial step allows us to move forward with ease.

Limit evaluation involves finding the value that a function approaches as the input gets close to a certain point.
In this problem, our target is to find \(\tlim_{x \rightarrow 0} \frac{\tsin^2 x}{2x^2}\).
We often use known limits to help in evaluations, such as \(\tlim_{x \to 0} \frac{\tsin x}{x} = 1\).
The trick is recognizing that \(\frac{\tsin^2 x}{2x^2}\) can be written as \(\frac{\tsin x}{x} \times \frac{\tsin x}{2x}\).
This breakdown allows us to individually evaluate the limits and multiply them:
\(\tlim_{x \to 0} \frac{\tsin x}{x} = 1\), so we have \(\frac{1}{2} \times 1 \times 1 = \frac{1}{2}\).
Thus, the limit of the given function is \(\frac{1}{2}\).

Simplifying expressions is crucial for solving limits and is often done to make the problem more manageable.
In the exercise, we simplify \(\frac{\tsin^2 x}{2x^2}\) by breaking it down:
First, we use the identity to convert \(\t1 - \tcos^2 x\) to \(\tsin^2 x\).
Next, we divide both the numerator and the denominator by \(\tx^2\), giving us \(\frac{\tsin x}{x} \times \frac{\tsin x}{2x}\).
Each part of this fraction is now easier to handle, especially since we know the limit \(\tlim_{x \to 0} \frac{\tsin x}{x} = 1\).
This simplification allows us to multiply these known limits together, yielding a final result of \(\frac{1}{2}\).
Mastering these techniques helps in tackling various trigonometric functions.