Pregunta
If \( 3 x-4 \leq f(x) \leq x^{2}-3 x+5 \) for \( x \geq 0 \), find \( \lim _{x \rightarrow 3} f(x) \)
Ask by Dickson Reese. in the United States
Feb 21,2025
Solución de inteligencia artificial de Upstudy
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The limit of f(x) as x approaches 3 is 5.
Solución
We are given the inequalities 3x - 4 ≤ f(x) ≤ x² - 3x + 5 for all x ≥ 0. To find limₓ→3 f(x), we first evaluate the lower and upper bound functions at x = 3.
For the lower bound:
3(3) - 4 = 9 - 4 = 5.
For the upper bound:
(3)² - 3(3) + 5 = 9 - 9 + 5 = 5.
Since both bounds are equal to 5 when x = 3, by the Squeeze Theorem, we conclude that:
limₓ→3 f(x) = 5.
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Simplifique esta solución Mind Expander
To find \( \lim _{x \rightarrow 3} f(x) \), we first need to evaluate the bounds given: 1. The lower bound is \( 3x - 4 \). 2. The upper bound is \( x^2 - 3x + 5 \). Let's calculate the limits of both bounds as \( x \) approaches 3. For the lower bound: \[ \lim_{x \rightarrow 3} (3x - 4) = 3(3) - 4 = 9 - 4 = 5. \] For the upper bound: \[ \lim_{x \rightarrow 3} (x^2 - 3x + 5) = (3)^2 - 3(3) + 5 = 9 - 9 + 5 = 5. \] Since both the lower and upper limits converge to the same value: \[ \lim_{x \rightarrow 3} f(x) = 5. \] Thus, the final answer is: \[ \lim_{x \rightarrow 3} f(x) = 5. \]
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