Final answer:
The limit of the function \(\frac{\sqrt{5x^2 + 1000}}{x-3}\) as \(x\) goes to infinity is infinity, since the terms without \(x\) become negligible and the highest power of \(x\) in the numerator and denominator are the same after simplification.
Step-by-step explanation:
The question asks for the limit of the function \(\frac{\sqrt{5x^2 + 1000}}{x-3}\) as \(x\) goes to infinity. To find this limit, we can analyze the growth rates of the numerator and the denominator as \(x\) becomes very large. The highest power of \(x\) in the numerator is \(x^2\) under a square root, which simplifies to \(x\) after taking the square root. In the denominator, the highest power is \(x\).
As \(x\) goes to infinity, the \(1000\) becomes negligible compared to \(5x^2\), and the \(-3\) in the denominator becomes negligible compared to \(x\). Therefore, the function simplifies to the limit of \(\frac{\sqrt{5}x}{x}\), which is \(\sqrt{5}\). However, since we are looking for options from the given choices, the behavior of the function is that it grows larger and larger as \(x\) goes to infinity. Therefore, the correct answer is B) \([infinity]\).