Evaluate \(\lim _{x \rightarrow \infty} \frac{2 x^{2}-x}{x^{2}+x}\) a) 2 b) 1 c) 0 d) \(-1\)

Calculus is a branch of mathematics focusing on limits, functions, integrals, and derivatives. It's indispensable for analyzing and understanding the changes between values that are related by a function. When it comes to limits, a fundamental concept in calculus, there's often a need to determine the behavior of a function as the input approaches a certain value, such as infinity.

In the exercise provided, we're tasked with evaluating the limit of a function as the variable approaches infinity. This requires understanding that the value of a limit isn’t always straightforward and one must often apply rules and techniques, such as factoring, canceling, or manipulating expressions, to find a limit that might not be immediately obvious. Particularly, we often look for the dominant term, which influences the function's behavior as the input becomes very large or very small.

Asymptotic behavior in calculus gives us a way to describe how functions behave as we approach large (or small) values for the variable, typically referred to as 'infinity' or 'negative infinity'. This idea is crucial when we deal with limits at infinity.

It’s like zooming out on a graph of the function and seeing which path it takes: Does it level off (horizontal asymptote), grow without bound, or oscillate? In rational functions, which are ratios of polynomials, if the degrees of the numerator and denominator are the same, the horizontal asymptote is found by dividing the leading coefficients, which is exactly what was done in the example.

Understanding asymptotic behavior allows students to predict the end behavior of a function and can be a valuable shortcut in calculating limits, especially when working with complex expressions.

Rational functions are ratios of polynomial functions, like the one given in the exercise, \(\frac{2x^2 - x}{x^2 + x}\). One interesting property of rational functions is that their behavior can be radically different at various parts of the number line. For example, they might have vertical asymptotes (where the function goes to infinity) or horizontal asymptotes, as we investigate when looking at the limit of a function as x approaches infinity.

To find the limit of a rational function as x approaches infinity, we observe the powers of x in the numerator and denominator. If the highest powers are the same, the limit will be the ratio of the coefficients of those terms. This was illustrated in the step-by-step solution by simplifying the function and finding that as x approaches infinity, the limit of the given function is 2.