Evaluate the limit
$\lim _{x \rightarrow \infty} \frac{1+6 x}{5-2 x}$
Answer (1)
Divide both numerator and denominator by x : lim x → ∞ x 5 − 2 x 1 + 6 .
As x approaches infinity, x 1 and x 5 approach 0.
Simplify the expression: 0 − 2 0 + 6 = − 2 6 .
Evaluate the limit: − 3 .
Explanation
Problem Analysis We are asked to find the limit of the rational function 5 − 2 x 1 + 6 x as x approaches infinity.
Strategy To evaluate this limit, we can divide both the numerator and the denominator by the highest power of x present, which in this case is x . This will help us simplify the expression and determine the limit as x goes to infinity.
Dividing by x Dividing both the numerator and the denominator by x , we get: x → ∞ lim x 5 − 2 x 1 + 6
Evaluating the Limit Now, as x approaches infinity, the terms x 1 and x 5 will approach 0. Therefore, the expression becomes: 0 − 2 0 + 6 = − 2 6 = − 3
Final Result Thus, the limit of the given function as x approaches infinity is -3.
Examples
In economics, this type of limit can be used to model the long-term behavior of cost-benefit ratios. For example, if 1 + 6 x represents the total benefit of a project and 5 − 2 x represents the total cost (where x is the scale of the project), finding the limit as x approaches infinity tells us the long-term benefit-cost ratio. This helps in making decisions about the sustainability and efficiency of large-scale projects. In this case, the limit being -3 suggests that as the scale increases indefinitely, the project becomes unsustainable, as costs outweigh benefits by a factor of 3.
