Find the limit, if it exists. (If an answer does not exist, enter DNE.)
[tex]$\lim _{x \rightarrow \infty} \frac{\sqrt{x^2-3}}{2 x-5}$[/tex]
Answer (1)
Divide both the numerator and the denominator by x to simplify the expression.
Rewrite the expression as lim x → ∞ 2 − x 5 1 − x 2 3 .
Evaluate the limit by noting that as x → ∞ , x 2 3 → 0 and x 5 → 0 .
The limit is 2 1 .
Explanation
Problem Analysis We are asked to find the limit of the function 2 x − 5 x 2 − 3 as x approaches infinity. To solve this, we need to analyze the behavior of the function as x becomes very large.
Dividing by x To find the limit, we can divide both the numerator and the denominator by x . This will help us simplify the expression and see what happens as x approaches infinity. Since we are considering x approaching infinity, we can assume 0"> x > 0 . Therefore, we can rewrite x x 2 − 3 as x 2 x 2 − 3 = 1 − x 2 3 .
Rewriting the Expression Now, we rewrite the original expression by dividing both the numerator and the denominator by x :
x → ∞ lim 2 x − 5 x 2 − 3 = x → ∞ lim x 2 x − 5 x x 2 − 3 = x → ∞ lim 2 − x 5 1 − x 2 3
Evaluating the Limit As x approaches infinity, the terms x 2 3 and x 5 approach 0. Therefore, we can evaluate the limit by substituting these values: x → ∞ lim 2 − x 5 1 − x 2 3 = 2 − 0 1 − 0 = 2 1 = 2 1 Thus, the limit of the function as x approaches infinity is 2 1 .
Final Answer The limit exists and is equal to 2 1 .
Examples
Imagine you are analyzing the efficiency of a manufacturing process where the cost per unit decreases as the number of units produced increases. The function 2 x − 5 x 2 − 3 could represent the cost per unit, where x is the number of units produced. Finding the limit as x approaches infinity tells you the minimum cost per unit you can achieve as production becomes very large. This helps in long-term planning and cost optimization.
