Question

Find the limits. $$ \lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+2 x}-x\right) $$

Short answer

The limit is 1.

Step-by-step solution

Understand the Expression

We need to evaluate the limit of the expression \( \sqrt{x^2 + 2x} - x \) as \( x \to \infty \). Notice that both terms inside the limit become infinitely large as \( x \) approaches infinity.

Rationalization

To simplify the expression and deal with indeterminate forms, multiply and divide by the conjugate: \( (\sqrt{x^2+2x} + x) \). This gives:\[\lim _{x \to \infty} \frac{(\sqrt{x^2+2x} - x)(\sqrt{x^2+2x} + x)}{\sqrt{x^2+2x} + x} = \lim _{x \to \infty} \frac{x^2 + 2x - x^2}{\sqrt{x^2+2x} + x}\]

Simplify the Expression

After expansion, the terms \(x^2\) cancel each other out, leaving \(2x\). The limit becomes:\[\lim_{x \to \infty} \frac{2x}{\sqrt{x^2+2x} + x}\]

Divide by Highest Power of \(x\)

To further simplify, divide the numerator and denominator by \(x\):\[\lim_{x \to \infty} \frac{2}{\sqrt{1 + \frac{2}{x}} + 1}\]

Evaluate the Limit

As \(x\) approaches infinity, \(\frac{2}{x}\) approaches zero. The expression simplifies to:\[\lim_{x \to \infty} \frac{2}{\sqrt{1} + 1} = \frac{2}{2} = 1\]

Conclusion: Limit Result

The limit of \( \sqrt{x^2 + 2x} - x \) as \( x \to \infty \) is 1.

Key concepts

Rationalization technique

Rationalization is a technique used in calculus to simplify expressions, especially when dealing with square roots. In many cases, simplifying square roots can help resolve complex indeterminate forms. Consider the expression \( \sqrt{x^2 + 2x} - x \). As \( x \) approaches infinity, both terms tend to grow infinitely, posing a challenge. By rationalizing, we multiply and divide the expression by its conjugate, \( \sqrt{x^2+2x} + x \), leading to the expression \( \lim_{x \to \infty} \frac{(\sqrt{x^2+2x} - x)(\sqrt{x^2+2x} + x)}{\sqrt{x^2+2x} + x} \).

This step aims to eliminate the troublesome square root. When multiplied out, the transformation simplifies the numerator by cancelling out terms, thereby giving a clearer form \( \frac{2x}{\sqrt{x^2+2x} + x} \), ensuring further simplification is possible.

Rationalization not only simplifies the expression but also brings about clarity when trying to tackle infinity limits by removing obstacles like radicals.

Indeterminate forms

In calculus, indeterminate forms appear commonly as expressions that do not straightforwardly reveal their limit as a variable approaches a certain value. Expressions like \( \frac{0}{0}, \infty - \infty \), or \( \frac{\infty}{\infty} \) are known as indeterminate forms.

Consider our example \( \sqrt{x^2 + 2x} - x \). As \( x \to \infty \), both terms tend toward infinity, creating an expression like \( \infty - \infty \). This is a classic indeterminate form where it is unclear what happens to the expression when both components grow boundlessly.

To solve such issues, further manipulations, like rationalization or applying L'Hôpital's rule, can be employed. The goal is to transform the expression into a form that reveals its limit. In this case, rationalization exposed a form simpler to evaluate, allowing us to eventually find the limit of one by avoiding separate infinite evaluations.

Infinity limits

Infinity limits relate to understanding how a function behaves as the variable grows beyond all finite bounds. When \( x \to \infty \), understanding whether functions approach a specific finite number, infinitely increase, or decrease becomes essential.

In the example \( \lim_{x \to \infty} (\sqrt{x^2 + 2x} - x) \), both \( \sqrt{x^2 + 2x} \) and \( x \) increase indefinitely. At first glance, it might seem challenging to predict what happens. Through simplification techniques like dividing by the highest power of \( x \) and rationalization, the problem becomes manageable.

Once rationalized and simplified, the expression transforms into \( \lim_{x \to \infty} \frac{2}{\sqrt{1 + \frac{2}{x}} + 1} \). As \( x \) increases, \( \frac{2}{x} \) approaches zero. Thus, the expression simplifies further to \( \frac{2}{2} = 1 \), a finite result. Understanding how to handle infinity limits helps in predicting the behavior of functions as they reach the vastness of infinity.