To solve this problem, we need to use the Squeeze Theorem and analyze the given functions to find the limit of $f(x)$ as $x \to 0$. The functions provided are:

• $u(x) = 1$
• $l(x) = 1 - x^2$
• $f(x) = \cos^2(x)$

1. Squeeze Theorem Concept:

The Squeeze Theorem states that if we have three functions $l(x) \leq f(x) \leq u(x)$ for all $x$ near a certain point (in this case, as $x \to 0$), and if both the upper bound $u(x)$ and lower bound $l(x)$ converge to the same limit, then $f(x)$ will also converge to that same limit.

2. Check the bounds at $x = 0$:

We need to check the limits of the upper and lower bounds as $x \to 0$.

• For $u(x) = 1$: $\lim_{x \to 0} u(x) = \lim_{x \to 0} 1 = 1$

• For ( l(x)