Limits: What Happens When a Function Approaches Infinity

A limit at infinity describes the value a function settles toward as x grows without bound, which is what determines a horizontal asymptote. Below: the definition, how to evaluate limits of rational, exponential and radical functions at infinity, and worked examples of each.

What is a Limit?

Before limits at infinity, it’s important to understand what a limit is. In simple terms, a limit examines what happens to a function \( f(x) \) as \( x \) approaches a certain value. Mathematically, this is denoted as:

\(lim_{{x \to a}} f(x) = L\)

This means that as \( x \) gets closer and closer to \( a \), the function \( f(x) \) gets closer and closer to the value \( L \).

Limits at Infinity

Limits at infinity are an extension of this concept. Instead of looking at what happens as \( x \) approaches a finite number, we explore what happens as \( x \) goes towards infinity \( \infty \) or negative infinity \( -\infty \).

This is expressed as:

\(lim_{{x \to \infty}} f(x) = L\)

or

\(lim_{{x \to -\infty}} f(x) = M\)

Here, \( L \) and \( M \) are the values that the function approaches as \( x \) becomes very large or very small, respectively.

Why are Limits at Infinity Important?

Limits at infinity are important because they give us a way to understand the end behavior of functions. This is fundamental in many areas of calculus, such as in studying asymptotes, evaluating improper integrals, and understanding the behavior of infinite series.

Examples

1. Polynomial Functions: Consider the function \( f(x) = x^2 \). As \( x \) approaches infinity, \( f(x) \) also approaches infinity. So, we write: \(lim_{{x \to \infty}} x^2 = \infty\)
2. Rational Functions: For \( f(x) = \frac{1}{x} \), as \( x \) approaches infinity, \( f(x) \) approaches \(0\). Therefore: \(\lim_{{x \to \infty}} \frac{1}{x} = 0\)

Conclusion

Understanding limits at infinity is a fundamental part of calculus and provides deep insights into the behavior of functions. Whether it’s a polynomial, rational, or any other type of function, analyzing its behavior as inputs become very large or very small is necessary for a thorough understanding of its overall characteristics.

FAQ

What are limits at infinity?

A limit at infinity is the value a function approaches as x grows without bound (x → ∞) or falls without bound (x → −∞). It describes the function’s end behavior. For example, as x gets larger and larger, 1/x shrinks toward 0, so the limit of 1/x as x → ∞ is 0. That result also tells you the graph of y = 1/x has a horizontal asymptote at y = 0.

What are the rules for limits at infinity?

For rational functions, compare the degrees of the numerator and denominator. If the top degree is smaller, the limit is 0. If the degrees are equal, the limit is the ratio of the leading coefficients. If the top degree is larger, the function grows toward ∞ or −∞. For example, (3x² + 1)/(2x² − 5) approaches 3/2 as x → ∞. Also remember that 1/xⁿ → 0 for any positive power n.

What is the definition of a limit in calculus?

The limit of f(x) as x approaches a equals L if f(x) gets as close to L as you want whenever x is close enough to a, without having to equal a. It is written lim x→a f(x) = L. For example, (x² − 4)/(x − 2) is undefined at x = 2, yet its limit there is 4, because the expression simplifies to x + 2 for every x other than 2.

What are some practice problems for limits at infinity?

Try these, then check your answers: (1) lim x→∞ of (5x − 2)/(x + 7) = 5, since the degrees match and 5/1 = 5. (2) lim x→∞ of (x + 1)/x² = 0, since the bottom has the higher degree. (3) lim x→∞ of x³/(x² + 1) = ∞, since the top has the higher degree. (4) lim x→∞ of e^(−x) = 0, because e^(−x) = 1/eˣ and eˣ grows without bound.

What is the limit of a trig function at infinity?

Functions like sin x and cos x have no limit as x → ∞, because they keep oscillating between −1 and 1 and never settle on one value. When a trig function is divided by something that grows, though, the limit can exist. For example, sin x / x → 0 as x → ∞, since the top stays between −1 and 1 while the bottom grows without bound (the Squeeze Theorem).