Evaluate the limit
$\lim _{x \rightarrow \infty} \frac{8 x+2}{2 x^2-5 x+9}$
Answer (1)
Divide both numerator and denominator by x 2 .
Simplify the expression to lim x → ∞ 2 − x 5 + x 2 9 x 8 + x 2 2 .
As x approaches infinity, terms with x in the denominator approach 0.
The limit evaluates to 2 0 = 0 .
Explanation
Problem Analysis We are asked to find the limit of the given rational function as x approaches infinity. The function is f ( x ) = 2 x 2 − 5 x + 9 8 x + 2 .
Strategy To evaluate the limit, we can divide both the numerator and the denominator by the highest power of x present in the denominator, which is x 2 . This will help us simplify the expression and determine the limit as x approaches infinity.
Dividing by x 2 Dividing both the numerator and the denominator by x 2 , we get: x → ∞ lim 2 x 2 − 5 x + 9 8 x + 2 = x → ∞ lim x 2 2 x 2 − x 2 5 x + x 2 9 x 2 8 x + x 2 2 = x → ∞ lim 2 − x 5 + x 2 9 x 8 + x 2 2
Evaluating the Limit As x approaches infinity, the terms x 8 , x 2 2 , x 5 , and x 2 9 all approach 0. Therefore, the limit becomes: x → ∞ lim 2 − x 5 + x 2 9 x 8 + x 2 2 = 2 − 0 + 0 0 + 0 = 2 0 = 0
Conclusion Thus, the limit of the given function as x approaches infinity is 0.
Examples
Imagine you are analyzing the long-term behavior of a business where the profit is modeled by a rational function like the one in this problem. As time (x) goes to infinity, you want to know if the profit will stabilize or approach a certain value. Evaluating the limit helps you understand the eventual trend of the profit, which is crucial for making strategic decisions. In this case, finding the limit as 0 suggests that the profit might diminish over a very long time, indicating a need for adjustments in the business model.
