Evaluate the limit [tex]$\lim _{x \rightarrow \infty} \frac{3 x^2-7 x+10}{11 x+9}$[/tex]
Answer (2)
Divide both the numerator and the denominator by x : lim x → ∞ 11 + x 9 3 x − 7 + x 10 .
As x approaches infinity, x 10 and x 9 approach 0.
The expression simplifies to lim x → ∞ 11 3 x − 7 .
As x approaches infinity, the limit is ∞ .
Explanation
Problem Analysis We are asked to evaluate the limit of a rational function as x approaches infinity. The given function is 11 x + 9 3 x 2 − 7 x + 10 . As x approaches infinity, the highest power terms dominate the behavior of the numerator and denominator.
Evaluating the Limit To evaluate the limit, we can divide both the numerator and the denominator by the highest power of x in the denominator, which is x . This gives us: x → ∞ lim 11 x + 9 3 x 2 − 7 x + 10 = x → ∞ lim x 11 x + x 9 x 3 x 2 − x 7 x + x 10 = x → ∞ lim 11 + x 9 3 x − 7 + x 10 As x approaches infinity, the terms x 10 and x 9 approach 0. Thus, we have: x → ∞ lim 11 + x 9 3 x − 7 + x 10 = x → ∞ lim 11 3 x − 7 As x approaches infinity, 3 x − 7 also approaches infinity. Therefore, the limit is: x → ∞ lim 11 3 x − 7 = ∞
Final Answer The limit of the given function as x approaches infinity is infinity.
Examples
In physics, when analyzing the motion of objects or the behavior of systems as time or distance becomes very large, evaluating limits at infinity helps to understand the long-term trends and stability of the system. For example, determining the terminal velocity of a falling object involves evaluating a limit as time approaches infinity. This concept is crucial for predicting the eventual state of a system and designing stable and efficient physical systems.
The limit of the given expression as x approaches infinity is ∞ . This is determined by simplifying the expression and recognizing that the higher degree term in the numerator dominates the behavior of the limit. Thus, the limit diverges to infinity.
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