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Lesson 3: Limits at Infinity and Asymptotes

Estimated time: 30-40 minutes

Learning Objectives

By the end of this lesson, you will be able to:

Evaluate limits as x approaches positive or negative infinity
Identify horizontal asymptotes using limits at infinity
Compare growth rates of polynomials to find limits of rational functions at infinity
Identify vertical asymptotes using infinite limits
Determine the end behavior of a function

Limits at Infinity

A limit at infinity describes the long-run behavior of a function as x grows without bound (positive or negative).

Limit at Infinity

We write lim_x→∞ f(x) = L if f(x) gets arbitrarily close to L as x increases without bound. Similarly for x → −∞.

Example 1: A Basic Limit at Infinity

Evaluate lim_x→∞ 1/x.

Solution: As x grows larger (10, 100, 1000, ...), 1/x gets smaller (0.1, 0.01, 0.001, ...). Therefore lim_x→∞ 1/x = 0.

Similarly, lim_x→−∞ 1/x = 0.

Key Fact

For any positive integer n: lim_x→∞ 1/xⁿ = 0 and lim_x→−∞ 1/xⁿ = 0.

Horizontal Asymptotes

If lim_x→∞ f(x) = L or lim_x→−∞ f(x) = L, then the line y = L is a horizontal asymptote of f. The graph of f approaches but never quite reaches this line.

Horizontal Asymptote

The line y = L is a horizontal asymptote of y = f(x) if lim_x→∞ f(x) = L or lim_x→−∞ f(x) = L (or both).

A function can have at most two horizontal asymptotes (one as x → ∞ and one as x → −∞), or fewer.

Limits of Rational Functions at Infinity

For a rational function p(x)/q(x), the limit at infinity depends on the degrees of the numerator and denominator. The technique is to divide every term by the highest power of x in the denominator.

Rational Function Rules at Infinity

Let f(x) = (a_nxⁿ + ...)/( b_mx^m + ...) where a_n, b_m ≠ 0.

If n < m (degree of numerator < degree of denominator): limit = 0
If n = m (degrees equal): limit = a_n/b_m (ratio of leading coefficients)
If n > m (degree of numerator > degree of denominator): limit = ±∞ (no horizontal asymptote)

Example 2: Degrees Equal

Evaluate lim_x→∞ (3x² + 5x − 1)/(7x² − 2).

Solution: Both numerator and denominator have degree 2. Divide every term by x²:

= (3 + 5/x − 1/x²)/(7 − 2/x²)

As x → ∞, the terms 5/x, 1/x², and 2/x² all → 0:

= (3 + 0 − 0)/(7 − 0) = 3/7

Horizontal asymptote: y = 3/7.

Example 3: Degree of Numerator Less Than Denominator

Evaluate lim_x→∞ (2x + 1)/(x³ − 4).

Solution: Degree 1 < degree 3, so the limit is 0.

Horizontal asymptote: y = 0.

Example 4: Degree of Numerator Greater Than Denominator

Evaluate lim_x→∞ (x³ + 2x)/(x − 1).

Solution: Degree 3 > degree 1. Divide: x³/x = x², which grows without bound.

lim_x→∞ = ∞. No horizontal asymptote.

Vertical Asymptotes

A vertical asymptote occurs at x = a when the function grows without bound near that point. This happens when the denominator approaches zero while the numerator does not.

Vertical Asymptote

The line x = a is a vertical asymptote of f if at least one of the following is true:

lim_x→a⁺ f(x) = ±∞
lim_x→a⁻ f(x) = ±∞

Example 5: Finding Vertical Asymptotes

Find the vertical asymptotes of f(x) = 1/(x² − 4).

Step 1: Factor the denominator: x² − 4 = (x − 2)(x + 2).

Step 2: The denominator is zero at x = 2 and x = −2.

Step 3: The numerator (1) is not zero at these points, so both x = 2 and x = −2 are vertical asymptotes.

Step 4: Check signs near x = 2:

As x → 2⁺: (x − 2) is small positive, (x + 2) ≈ 4, so f(x) → +∞
As x → 2⁻: (x − 2) is small negative, (x + 2) ≈ 4, so f(x) → −∞

Limits at Infinity for Non-Rational Functions

Example 6: Exponential Function

Evaluate lim_x→∞ e^−x.

Solution: As x → ∞, −x → −∞, so e^−x → 0. Therefore the limit is 0.

Horizontal asymptote: y = 0 (to the right).

Example 7: Square Root at Infinity

Evaluate lim_x→∞ (x − √(x² + 3x)).

Solution: This is an ∞ − ∞ indeterminate form. Rationalize:

Multiply by (x + √(x² + 3x))/(x + √(x² + 3x)):

= [x² − (x² + 3x)] / [x + √(x² + 3x)] = −3x / [x + √(x² + 3x)]

Divide top and bottom by x: = −3 / [1 + √(1 + 3/x)]

As x → ∞: = −3 / [1 + √1] = −3/2 = −3/2

End Behavior Summary

Knowing limits at infinity tells us the "end behavior" of a function — what happens as you move far to the left or far to the right on the graph.

Polynomials: End behavior determined by the leading term (highest-degree term).
Rational functions: End behavior determined by the degree comparison rule above.
Exponentials: e^x → ∞ as x → ∞ and e^x → 0 as x → −∞.
Logarithms: ln x → ∞ as x → ∞, but grows more slowly than any positive power of x.

Key Takeaways

Limits at infinity describe long-run behavior; if the limit is L, the line y = L is a horizontal asymptote.
For rational functions, compare the degrees: bottom wins → 0, same degree → ratio of leading coefficients, top wins → ∞.
The technique of dividing by the highest power of x works for all rational functions at infinity.
Vertical asymptotes occur where the denominator is zero and the numerator is not (after simplification).
A function can have at most two horizontal asymptotes but any number of vertical asymptotes.

Check Your Understanding

1. Find lim_x→∞ (5x³ − 2x)/(4x³ + x − 7).

Answer: Degrees are equal (both 3). The limit equals the ratio of leading coefficients: 5/4. Horizontal asymptote: y = 5/4.

2. Find the horizontal asymptote(s) of f(x) = (x + 1)/(x² + 1).

Answer: Degree of numerator (1) < degree of denominator (2), so lim_x→±∞ f(x) = 0. Horizontal asymptote: y = 0.

3. Find all vertical asymptotes of g(x) = (x + 3)/[(x − 1)(x + 5)].

Answer: The denominator is zero at x = 1 and x = −5. The numerator is not zero at either point. So the vertical asymptotes are x = 1 and x = −5.

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