Question

Find the limit. \(\lim _{x \rightarrow \infty}\left(2 x-x^{-2}\right)\)

Short answer

The limit as x approaches infinity of the function \(2x - x^{-2}\) is infinity.

Step-by-step solution

Identify Dominant Term

The given function is \(2x - x^{-2}\). As x approaches infinity, the term with the highest power in x will dominate the function. In this case, \(2x\) is the term with the highest power, as it is directly proportional to x. So, while \(x^{-2}\) will also go to infinity, it will do so at a far slower rate and can be considered insignificant compared to \(2x\).

Figure Out the Limit

Considering the conclusion from step 1, the term \(x^{-2}\) becomes irrelevant as \(x\) approaches infinity, and the limit will be dictated by the term \(2x\). Hence, as x approaches infinity, the limit of the function \(2x - x^{-2}\) will be equivalent to the limit of the function \(2x\), which is also infinity. Thus, the limit of this function as x approaches infinity is infinity.

Key concepts

Calculus

Calculus is a branch of mathematics focused on the study of change and motion. Within calculus, limits are fundamental in exploring how functions behave as inputs approach a certain value.

In our exercise, we use calculus to examine what happens to the function \[ 2x - x^{-2} \] as \( x \), the input, grows without bound. This kind of problem is a classic example encountered when learning about limits in calculus. It helps us to understand how functions act near certain points, or as they extend towards infinity. This knowledge is crucial when we're trying to find the slope of a curve (derivative) or the area under it (integral).

Asymptotic Behavior

Asymptotic behavior in calculus describes how a function acts as the input either grows very large (heads towards infinity) or becomes very small (approaches zero).

The exercise \[ \lim _{x \rightarrow \infty}(2 x-x^{-2}) \] probes exactly this—the function's behavior at the 'asymptotes' or as \( x \) tends toward infinity. By examining a function's asymptotic behavior, we can anticipate the function's end behavior and any horizontal or vertical asymptotes, which are lines that the graph of a function approaches but never crosses.

Dominant Terms

The concept of dominant terms is illustrated when we analyze the given function. In \( 2x - x^{-2} \), \( 2x \) becomes the dominant term as \( x \) approaches infinity because it grows much faster than the \( x^{-2} \) term due to its higher degree (1 versus -2).

When determining limits, especially when \( x \) tends toward infinity, the dominant term dictates the end behavior of the function. Lesser terms, especially those with negative powers like \( x^{-2} \) in our case, tend to zero and barely affect the limit. Hence, in problems involving limits at infinity, identifying and focusing on the dominant term simplifies the process significantly.

Infinity in Limits

Dealing with infinity in limits is about understanding behavior beyond bounds. In calculus, we don't think of infinity as a number, but rather as a concept that describes something without limit.

When we calculate \[ \lim _{x \rightarrow \infty}(2 x-x^{-2}) \], we interpret the \( \infty \) symbol to mean that \( x \) is growing without bound. This is different from a limit where \( x \) approaches a finite number. Instead, we’re interested in where the function is headed—its ultimate behavior—and not a specific value. For the function at hand, as \( x \) becomes very large, the \( 2x \) term dominates and drives the function towards infinity, which is the limit. Understanding this concept is vital for grasping the big picture in many areas of calculus, including series convergence and improper integrals.