The image shows a mathematical problem related to the Squeeze Theorem (Teorema Squeeze in Indonesian) for limits. Here's the problem breakdown:

You are asked to sketch three functions:

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

Then, you are instructed to use these graphs and apply the Squeeze Theorem to determine the limit:

$\lim_{x \to 0} f(x)$

Applying the Squeeze Theorem

The Squeeze Theorem states that if: $l(x) \leq f(x) \leq u(x)$ for all $x$ near some point (except possibly at the point itself), and if: $\lim_{x \to 0} l(x) = \lim_{x \to 0} u(x) = L,$ then: $\lim_{x \to 0} f(x) = L.$

Step-by-Step:

1. Upper Bound: $u(x) = 1$ is constant, and $\lim_{x \to 0} u(x) = 1$.

2. Lower Bound: $l(x) = 1 - x^2$ approaches 1 as $x \to 0$, i.e., $\lim_{x \to 0} l(x) = 1$.

3. Function: $f(x) = \cos^2(x)$. At $x = 0$, $\cos^2(0) = 1$, so $\lim_{x \to 0} f(x) = 1$.

Since both the upper and lower bounds converge to 1 as $x \to 0$, by the Squeeze Theorem: $\lim_{x \to 0} \cos^2(x) = 1.$

Conclusion:

$\lim_{x \to 0} f(x) = 1.$

Would you like to see more details on the graph or how to apply the Squeeze Theorem in other scenarios? Here are a few related questions to deepen your understanding:

1. Can you visualize the graphs of $u(x)$, $l(x)$, and $f(x)$ to confirm the bounds?
2. How does the Squeeze Theorem apply when the limits are not equal at a point?
3. Can we use the Squeeze Theorem for limits approaching infinity?
4. What happens if $f(x)$ does not lie between $u(x)$ and $l(x)$?
5. How would the result change if we were considering $\lim_{x \to \infty} f(x)$?

Tip: The Squeeze Theorem is especially useful for functions that are difficult to evaluate directly but can be bounded by simpler ones.