Introduction

One of the most characteristic elements of calculus is infinity and the limit at infinity. The concept of infinity does not point to a real number; instead, it attempts to describe a magnitude that exceeds any real bound. For example, when we have the function f(x) = 1/x and we inquire about its behavior when x becomes as large as we want, as x tends to infinity (x\to \infty), we observe that f(x) can approach zero as closely as desired. Thus, we write:

\displaystyle \lim_{x\to + \infty}\dfrac{1}{x} = 0

Graphically, this situation looks as follows:

Definition of Limit at Infinity

From this idea we just introduced, we can formulate the mathematical definition of limit at infinity:

\displaystyle \lim_{x\to +\infty}f(x) = L := (\forall\epsilon\gt 0) (\exists M\in\mathbb{R})(M\lt x \rightarrow |f(x) - L|\lt \epsilon )

\displaystyle \lim_{x\to -\infty}f(x) = L := (\forall\epsilon\gt 0) (\exists N\in\mathbb{R})(x\lt N \rightarrow |f(x) - L|\lt \epsilon )

The intuitive notion of this limit indicates what happens with f(x) when x moves as far away from the origin as we want, either to the right or left. The strategy for calculating limits at infinity is not very different from the one we use for finite limits because their algebra is practically the same, we just need to consider the following results:

Basic Limits at Infinity

Based on these definitions, we can prove the following basic limits.

1. \displaystyle \lim_{x\to \pm\infty}k = k
2. \displaystyle \lim_{x\to \pm\infty}\dfrac{1}{x} = 0

PROOF:

1. By the definition of limit at infinity, we have that \displaystyle \lim_{x\to +\infty}k = k is equivalent to saying: (\forall\epsilon\gt 0) (\exists M\in\mathbb{R})\left(M\lt x \rightarrow \left|k-k\right|\lt \epsilon \right). But \left|k-k\right|=0\lt \epsilon always holds for any \epsilon \gt 0, and regardless of the value of M, so the limit is assured.

    

2. We know that by definition \displaystyle \lim_{x\to +\infty}k = k is equivalent to saying: (\forall\epsilon\gt 0) (\exists M\in\mathbb{R})\left(M\lt x \rightarrow \left|\dfrac{1}{x}\right|\lt \epsilon \right). But this implication is immediately satisfied if we consider M=1/\epsilon, so the limit is assured.

    

These proofs are carried out analogously for when x\to+\infty.

Algebra of Limits at Infinity

The algebra of infinite limits is analogous to that of finite limits. If \displaystyle \lim_{x\to \pm \infty}f(x) = L and \displaystyle \lim_{x\to \pm \infty}g(x) = M, then the following rules apply:

1. Sum and Subtraction of Limits: \displaystyle \lim_{x\to \pm\infty}(f(x)\pm g(x)) = L \pm M
2. Multiplication by a constant: \displaystyle \lim_{x\to \pm\infty}cf(x) = cL
3. Product of Limits: \displaystyle \lim_{x\to \pm\infty}f(x)g(x) = LM
4. Division of Limits: As long as M\neq 0, then \displaystyle \lim_{x\to \pm\infty}f(x)/g(x)=L/M
5. Powers of Limits: If p,q \in\mathbb{Z} and q\neq 0, then \displaystyle \lim_{x\to \pm\infty}[f(x)]^{p/q} = L^{p/q}. If q is even, it is assumed that L\geq 0

In fact, the proof of all these properties is analogous to that of finite limits

Limit at Infinity in Rational Functions

A rational function is one that can be expressed as a quotient of two polynomials. When calculating limits at infinity for this type of functions, we can observe a very useful property:

Let’s suppose we want to calculate \displaystyle \lim_{x\to \infty}P(x)/Q(x)

• If the degree of P(x) is greater than that of Q(x), then the size of the function f(x) will grow without limit as x\to\infty (the limit will not exist).
• If the degree of P(x) is less than that of Q(x), then the limit will be zero.
• And finally, if the degree of P(x) is equal to that of Q(x), then the limit will be equal to the quotient of the coefficients of the highest degree term.

The best part of this result is that, as we will see in the following examples, it works similarly even if the powers involved are not integers.

Examples of Limits at Infinity

1. \displaystyle \lim_{x\to +\infty}\dfrac{x+1}{x^2+3} [SOLUTION]
2. \displaystyle \lim_{x\to -\infty}\dfrac{2x^3 + 7}{x^3 - x^2 + x + 7} [SOLUTION]
3. \displaystyle \lim_{x\to +\infty}\dfrac{9x^4 + x}{2x^4 + 5x^2 - x + 6} [SOLUTION]
4. \displaystyle \lim_{x\to +\infty}\dfrac{10x^5 + x4 + 31}{x^4 - 7x^3 + 7x^2 + 9} [SOLUTION]
5. \displaystyle \lim_{x\to +\infty}\dfrac{2\sqrt{x}+x^{-1}}{3x - 7} [SOLUTION]
6. \displaystyle \lim_{x\to -\infty}\dfrac{2x^{5/3} - x^{1/3} + 7}{x^{8/5}+3x + \sqrt{x}} [SOLUTION]
7. \displaystyle \lim_{x\to +\infty}\dfrac{\sqrt[3]{x}-5x+3}{2x + x^{2/3} - 4} [SOLUTION]
8. \displaystyle \lim_{x\to +\infty}\dfrac{x^{8/3}+2x + \sqrt{x}}{x^2+x-3} [SOLUTION]