https://file.aiquickdraw.com/mathbot/file/172551320826143rnz8mr.jpg

The given problem is:

\[
1 - \frac{x^2}{4} \leq u(x) \leq 1 + \frac{x^2}{2}, \text{ for all } x \neq 0,
\]
and we are asked to find:

\[
\lim_{{x \to 0}} u(x).
\]

### Applying the Squeeze Theorem

To find the limit, we can apply the **Squeeze Theorem** (also known as the Sandwich Theorem). The Squeeze Theorem states that if \( f(x) \leq g(x) \leq h(x) \) for all \( x \) in some interval around \( a \), except possibly at \( a \) itself, and if 

\[
\lim_{{x \to a}} f(x) = \lim_{{x \to a}} h(x) = L,
\]
then 

\[
\lim_{{x \to a}} g(x) = L.
\]

In our case:

\[
f(x) = 1 - \frac{x^2}{4}, \quad g(x) = u(x), \quad h(x) = 1 + \frac{x^2}{2}.
\]

Now, let's find the limits of \( f(x) \) and \( h(x) \) as \( x \to 0 \).

\[
\lim_{{x \to 0}} \left(1 - \frac{x^2}{4}\right) = 1 - 0 = 1,
\]
\[
\lim_{{x \to 0}} \left(1 + \frac{x^2}{2}\right) = 1 + 0 = 1.
\]

Since both limits are equal to 1, we can apply the Squeeze Theorem:

\[
\lim_{{x \to 0}} u(x) = 1.
\]

### Conclusion

The limit of \( u(x) \) as \( x \to 0 \) is:

\[
\lim_{{x \to 0}} u(x) = 1.
\]

Would you like more details on this solution or have any questions?

### Related Questions to Explore:
1. How does the Squeeze Theorem work for trigonometric functions?
2. What are some other applications of the Squeeze Theorem?
3. Can the Squeeze Theorem be used for complex functions?
4. What happens if one of the bounds diverges to infinity?
5. How would this problem change if we approached a different point?

### Tip:
When using the Squeeze Theorem, ensure that both bounding functions have the same limit at the point of interest!

Given that 1 - (x^2 / 4) ≤ u(x) ≤ 1 + (x^2 / 2) for all x ≠ 0, find lim(x→0) u(x), no matter how complicated u is.

Solve a limit problem using the Squeeze Theorem. Given 1 - (x^2 / 4) ≤ u(x) ≤ 1 + (x^2 / 2) for all x ≠ 0, find lim(x→0) u(x). Learn how to apply the Squeeze Theorem to compute the limit step-by-step.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation