Ch. 2 - Limits and Continuity

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 2 - Limits and Continuity

Problema 75

Capitolo 2, Problema 75

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?

Guida verificata passo dopo passo

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 and 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 and 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.

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Limit of a Function

The limit of a function describes the value that a function approaches as the input approaches a certain point. In this context, understanding limits is crucial for determining the behavior of f(x) as x approaches specific points, particularly where the function is bounded by x² and x⁴.

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Limits of Rational Functions: Denominator = 0

Squeeze Theorem

The Squeeze Theorem states that if a function is squeezed between two other functions that have the same limit at a point, then the squeezed function must also approach that limit. This theorem is applicable here since f(x) is bounded by x² and x⁴, allowing us to infer limits at points where these bounding functions converge.

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Percorso guidato

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Fundamental Theorem of Calculus Part 1

Continuity of Functions

A function is continuous at a point if the limit as x approaches that point equals the function's value at that point. In this problem, identifying points of continuity for f(x) is essential, as it helps determine where the limit exists and what its value is, particularly at the boundaries of the intervals given.

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Intro to Continuity

Pratica correlata

Domanda del libro di testo

Find the limits in Exercises 59–62. Write ∞ or −∞ where appropriate.

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Oblique Asymptotes

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y = x² / (x − 1)

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Theory and Examples

Suppose that g(x) ≤ f(x) ≤ h(x) for all x≠2 and suppose that lim x→2 g(x) = lim x→2 h(x) = −5. Can we conclude anything about the values of f, g, and h at x = 2? Could f(2) = 0? Could limx→2 f(x)=0? Give reasons for your answers.

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Use the formal definitions from Exercise 97 to prove the limit statements in Exercises 98–102.

lim x→2⁻ (1 / (x − 2)) = −∞

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Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.

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d. x→2

207