Evaluate the limit [tex]$\lim _{x \rightarrow \infty} \frac{9 x^3-2 x^2-3 x}{8-11 x-3 x^3}$[/tex]
Answer (2)
Divide both the numerator and the denominator by x 3 to simplify the expression.
Rewrite the limit as lim x → ∞ x 3 8 − x 2 11 − 3 9 − x 2 − x 2 3 .
Evaluate the limit by considering the behavior of each term as x approaches infinity, where terms of the form x n c approach 0.
Simplify the expression to find the final value of the limit: − 3 .
Explanation
Analyze the limit We are asked to evaluate the limit of a rational function as x approaches infinity: x → ∞ lim 8 − 11 x − 3 x 3 9 x 3 − 2 x 2 − 3 x To find this limit, we can divide both the numerator and the denominator by the highest power of x present in the denominator, which is x 3 . This will help us simplify the expression and determine the limit as x goes to infinity.
Divide by highest power Divide both the numerator and the denominator by x 3 :
x → ∞ lim x 3 8 − x 3 11 x − x 3 3 x 3 x 3 9 x 3 − x 3 2 x 2 − x 3 3 x This simplifies to: x → ∞ lim x 3 8 − x 2 11 − 3 9 − x 2 − x 2 3
Evaluate the limit Now, as x approaches infinity, the terms x 2 , x 2 3 , x 3 8 , and x 2 11 all approach 0. Therefore, the limit becomes: x → ∞ lim 0 − 0 − 3 9 − 0 − 0 = − 3 9
Final result Finally, simplify the expression: − 3 9 = − 3 Thus, the limit is -3.
Examples
In electrical engineering, when analyzing circuits with complex impedances, you might encounter rational functions similar to the one in this problem. Determining the limit as the frequency approaches infinity helps understand the circuit's behavior at very high frequencies, which is crucial for designing filters and ensuring signal integrity. By evaluating such limits, engineers can predict the circuit's stability and performance under extreme conditions.
The limit lim x → ∞ 8 − 11 x − 3 x 3 9 x 3 − 2 x 2 − 3 x simplifies to − 3 by dividing by the highest power of x and evaluating as x approaches infinity. The final result is − 3 .
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