Question

Theory and ExamplesIf x⁴ ≤ f(x) ≤ x² for x in [−1,1] and x² ≤ f(x) ≤ x⁴ for x  1, at what points c do you automatically know limx→c f(x)? What can you say about the value of the limit at these points?

Accepted Answer

First, understand the problem statement: We have a function f(x) that is bounded by x⁴ and x² in different intervals. We need to find points c where the limit of f(x) as x approaches c is known. Consider the interval x ∈ [−1,1]. Here, the function f(x) is bounded by x⁴ ≤ f(x) ≤ x². Since both x⁴ and x² are continuous functions, and f(x) is squeezed between them, we can apply the Squeeze Theorem. The Squeeze Theorem states that if g(x) ≤ f(x) ≤ h(x) and limx→c g(x) = limx→c h(x) = L, then limx→c f(x) = L. In the interval [−1,1], both x⁴ and x² approach the same value as x approaches any point c within this interval. Now, consider the intervals x  1. Here, f(x) is bounded by x² ≤ f(x) ≤ x⁴. Again, both bounding functions are continuous, and the Squeeze Theorem can be applied similarly. For points c within the intervals x  1, the limits of x² and x⁴ as x approaches c are equal, allowing us to conclude that limx→c f(x) is known at these points. The value of the limit at these points is the same as the limit of the bounding functions.

Theory and Examples

If x⁴ ≤ f(x) ≤ x² for x in [−1,1] and x² ≤ f(x) ≤ x⁴ for x < - 1 and x > 1, at what points c do you automatically know limx→c f(x)? What can you say about the value of the limit at these points?

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Key Concepts

Limit of a Function

Squeeze Theorem

Continuity of Functions

Theory and ExamplesIf x⁴ ≤ f(x) ≤ x² for x in [−1,1] and x² ≤ f(x) ≤ x⁴ for x < - 1 and x > 1, at what points c do you automatically know limx→c f(x)? What can you say about the value of the limit at these points?