Math Problem Statement
Solution
The given problem is:
and we are asked to find:
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 for all in some interval around , except possibly at itself, and if
then
In our case:
Now, let's find the limits of and as .
Since both limits are equal to 1, we can apply the Squeeze Theorem:
Conclusion
The limit of as is:
Would you like more details on this solution or have any questions?
Related Questions to Explore:
- How does the Squeeze Theorem work for trigonometric functions?
- What are some other applications of the Squeeze Theorem?
- Can the Squeeze Theorem be used for complex functions?
- What happens if one of the bounds diverges to infinity?
- 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!
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Math Problem Analysis
Mathematical Concepts
Limits
Squeeze Theorem
Inequalities
Formulas
Squeeze Theorem: If f(x) ≤ g(x) ≤ h(x) and lim(x→a) f(x) = lim(x→a) h(x) = L, then lim(x→a) g(x) = L
Limit of f(x) = 1 - (x^2 / 4) as x → 0 is 1
Limit of h(x) = 1 + (x^2 / 2) as x → 0 is 1
Theorems
Squeeze Theorem
Suitable Grade Level
Grades 11-12 or College Level