Limits at Infinity

Limits are a basic principle in calculus which explains the behavior of functions as they approach a specific value or as the variable approaches infinity. Limits at infinity are a particular type of limits which deals with the way functions behave when the input variable approaches either positive or negative infinity. Knowing the limit at infinity is a significant factor in the analysis of the asymptotic behavior of functions and their properties, which is the basis for different areas in mathematics, science and engineering.

The function f(x) limit as x approaches a particular value a is written as lim_(x→a) f(x). It stands for the value that f(x) gets when x is near a, but not necessarily equal to f(a). To give you an instance let us take f(x) = 1/x. While x is getting closer and closer to 0 from the right, f(x) is getting larger, and while x is getting closer and closer to 0 from the left, f(x) is getting smaller. However, f(0) is undefined. In this case, we say that lim_(x→0) f(x) does not exist.

The limit at infinity is the limit that describes the behavior of the function as x approaches plus or minus infinity. The limit of a function f(x) that is approached as x goes towards positive infinity is called lim_(x→∞) f(x), and the term of the function as x goes towards negative infinity is called lim_(x→-∞) f(x). The limits are limits that are the ones that demonstrate the asymptotic behavior of the function, which are the values that f(x) is getting close to as x becomes very large or very small.

Formal Definition of Limits at Infinity

The formal definition of limits at infinity states that lim_(x→∞) f(x) = L if for every ε > 0, there exists a real number M such that |f(x) – L| < ε for all x > M. Similarly, lim_(x→-∞) f(x) = L if for every ε > 0, there exists a real number M such that |f(x) – L| < ε for all x < -M.

Representation of Limits at Infinity

Limits at infinity can be shown through a sketch of the graph of the function and by the observation of its behavior as x goes to positive or negative infinity. As well, asymptotes are used in the representation of the limits at infinity. The horizontal asymptote is a horizontal line that the graph of the function gets near the x is going to the positive or negative infinity.

The properties of limit at infinity are mentioned below:

Constant Rule: If f(x) = c, where c is a constant, then lim_(x→∞) f(x) = c and lim_(x→-∞) f(x) = c.

Power Rule: If f(x) = xⁿ, where n is a positive integer, then:

• If n is even, lim_(x→∞) f(x) = ∞ and lim_(x→-∞) f(x) = ∞.
• If n is odd, lim_(x→∞) f(x) = ∞ and lim_(x→-∞) f(x) = -∞.

Reciprocal Rule: If f(x) = 1/x, then lim_(x→∞) f(x) = 0 and lim_(x→-∞) f(x) = 0.

Sum Rule: If lim_(x→∞) f(x) = L1 and lim_(x→∞) g(x) = L2, then lim_(x→∞) [f(x) + g(x)] = L1 + L2.

Difference Rule: If lim_(x→∞) f(x) = L1 and lim_(x→∞) g(x) = L2, then lim(x→∞) [f(x) – g(x)] = L1 – L2.

Product Rule: If lim_(x→∞) f(x) = L1 and lim_(x→∞) g(x) = L2, then lim_(x→∞) [f(x) * g(x)] = L1 * L2.

Quotient Rule: If lim_(x→∞) f(x) = L1 and lim_(x→∞) g(x) = L2 ≠ 0, then lim_(x→∞) [f(x) / g(x)] = L1 / L2.

To determine the limits at infinity, we can utilize several methods, for instance, direct substitution, factoring, rationalizing the numerator or the denominator, and the properties of limits. Let’s consider an example:

Example: Evaluate lim_(x→∞) (x² + 3x) / (2x² – 5).

Solution:

Divide the numerator and denominator by the highest power of x in the denominator, which is x².

(x² + 3x) / (2x² – 5) = (1 + 3/x) / (2 – 5/x²)

As x approaches infinity, 3/x approaches 0 and 5/x² approaches 0.

Therefore, lim_(x→∞) (x² + 3x) / (2x² – 5) = lim_(x→∞) (1 + 3/x) / (2 – 5/x²) = 1/2.

Limits at infinity are the main point in calculus, especially in the area of asymptotic behavior, continuity, and differentiability of functions. They are employed to study the mode of functions as they come to close or far from infinity and to check the presence of horizontal asymptotes.

Limits of Rational Functions at Infinity

The limit of a rational function f(x) = P(x) / Q(x) as x approaches infinity depends on the degrees of the polynomials P(x) and Q(x). If deg(P(x)) < deg(Q(x)), then lim_(x→∞) f(x) = 0. If deg(P(x)) > deg(Q(x)), then lim_(x→∞) f(x) = ∞. If deg(P(x)) = deg(Q(x)), then lim_(x→∞) f(x) = a, where a is the ratio of the leading coefficients of P(x) and Q(x).

For example, consider the rational function f(x) = (x² + 3x + 1) / (2x² – 5). Here, deg(P(x)) = 2 and deg(Q(x)) = 2, so lim_(x→∞) f(x) = 1/2.

Limits of Trigonometric Functions at Infinity

The limits of the trigonometric functions when x goes to infinity are determined by the type of function and its periodicity. To illustrate this, lim_(x→∞) sin(x) and lim_(x→∞) cos(x) do not exist because the functions oscillate between -1 and 1 without getting closer to any specific value. On the other hand, lim_(x→∞) sin(x) / x = 0 and lim_(x→∞) cos(x) / x = 0.

Limits of Exponential and Logarithmic Functions at Infinity

Exponential functions of the form f(x) = a^x, where a > 0, have the following limits:

• If a > 1, then lim_(x→∞) f(x) = ∞ and lim_(x→-∞) f(x) = 0.
• If 0 < a < 1, then lim_(x→∞) f(x) = 0 and lim_(x→-∞) f(x) = ∞.

Logarithmic functions of the form f(x) = log_a(x), where a > 0 and a ≠ 1, have the following limits:

If x approaches ∞, then f(x) approaches ∞.

If x approaches 0+ (positive values), then f(x) approaches -∞.

The limits at infinity are a basic idea in calculus which explains the behavior of functions as they approach either positive or negative infinity. The limits at infinity are very important when one wants to know the asymptotic behavior of functions as well as their properties, which is necessary to use in many mathematics, science, and engineering areas. Through the use of the given properties of limits and the application of techniques, such as direct substitution, factoring, and rationalizing, we can find the limit at infinity and thus determine the behavior of functions when they are approaching infinity. The research of limits at infinity is also important in the calculus of rational, trigonometric, exponential, and logarithmic functions, as it brings certain behavior of their asymptotes and the existence of horizontal asymptotes to light.

Related Articles

• Limits in Calculus
• Formal Definition of Limits
• Properties of Limits

Example 1: Evaluate the limit as x approaches infinity for the function ?(?) = 2x³−5x²+3

Solution:

As x approaches infinity, the term with the highest power of x dominates the function. Therefore, the limit is:

lim⁡_?→∞(2x³−5x²+3) = ∞

Example 2: Find the limit as x approaches infinity for the function ?(?)=4x²+2 / x³−3

Solution:

Divide the numerator and denominator by the highest power of x to simplify the expression:

lim⁡_?→∞4x²+2x³−3 = lim⁡_?→∞ 4/x +2/x² / 1−3/x³

As x approaches infinity, the terms with 1/? and 1/?² approach 0, leading to:

lim_⁡?→∞ 4x²+2x³−3 = 0+0 / 1−0 = 0

Q1. Evaluate the limit as x approaches infinity for the function f(x) = 3x² − 2x + 5

Q2. Compute the limit as x approaches infinity for the function f(x) = 5x²+3x−12/x³+7

Q3. Calculate the limit as x tends to infinity for the function f(x) = e^-x

Q4. Evaluate the limit as x tends to negative infinity for the function f(x) = e^2x -x²

What is a limit at infinity?

A limit at infinity describes the behavior of a function as the input approaches positive or negative infinity.

What does the limit equal?

The limit at infinity equals the value that the function approaches as the input grows infinitely large or small.

What functions approach infinity?

Functions like polynomials with higher-degree terms, exponential functions with positive bases, and rational functions with higher-degree numerators and denominators tend to approach infinity.

What are the rules of infinity?

Rules for limits at infinity include the dominance rule (terms with higher degrees dominate), the constant multiple rules, and the sum/difference rule.

What functions approach negative infinity?

Functions like negative exponential functions, rational functions with higher-degree terms in the denominator, and functions involving negative powers tend to approach negative infinity.