Limits at Infinity

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).

Limits at Infinity

The limit at infinity is the limit that describes the behavior of the function as x approaches plus or minus infinity.

• 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.
• 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.

Properties of Limits at 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.

Evaluating Limits at Infinity

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.

Calculus of Limits at Infinity

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 -∞.

Related Articles

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

Solved Examples in Limits at infinity

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

Practice Questions on Limits at Infinity

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