1.15 - Connecting Limits at Infinity and Horizontal Asymptotes

Introduction to Horizontal Asymptotes

A horizontal asymptote describes the end behavior of a function, meaning what happens to the

y

-values as

x

heads toward positive or negative infinity.

Key Difference from Vertical Asymptotes

Horizontal asymptotes:

Describe behavior as $x \to \pm\infty$

The graph CAN cross the asymptote

Represent the value the function approaches in the long run

Vertical asymptotes:

Occur at specific $x$ -values where the function is undefined

The graph CANNOT cross the asymptote

The function approaches $\pm\infty$ near the asymptote

Visual Understanding

1.15 - Connecting Limits at Infinity and Horizontal Asymptotes

x \to \infty

, the function approaches

y = 2

. Notice the graph crosses the asymptote several times, which is totally allowed for horizontal asymptotes.

Limit Notation for End Behavior

Horizontal asymptotes get written with limits.

Right-end behavior (as $x$ goes to positive infinity):

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

Left-end behavior (as $x$ goes to negative infinity):

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

If either limit equals a finite number

L

, then

y = L

is a horizontal asymptote.

Growth Rates of Functions

To find horizontal asymptotes of rational functions, you need to know which types of functions grow fastest.

Ranking by Growth Rate (Fastest to Slowest)

Think about very large values of

x

, like

x = 1000

1. Exponential functions (fastest)

Examples: $4^x$ , $e^x$ , $2^x$

Growth: extremely rapid

2. Polynomial functions (moderate)

Examples: $x^{60}$ , $x^3$ , $x^2$

Growth: depends on the degree (higher degree means faster)

3. Logarithmic functions (slowest)

Examples: $\ln(x)$ , $\log(x)$

Growth: very slow, but it never levels off. $\ln x$ keeps growing without bound, so it has no horizontal asymptote

Comparison Table

$x$	$\ln(x)$	$x^2$	$x^{10}$	$2^x$
10	2.3	100	$10^{10}$	1024
100	4.6	10,000	$10^{20}$	$1.27 \times 10^{30}$

Key insight: for large

x

, exponential functions dominate everything else.

Comparing Growth with Limits

To compare how fast two functions grow, take the limit of their ratio as

x \to \infty

If $\lim_{x \to \infty} \frac{f(x)}{g(x)} = \infty$ , then $f$ grows faster than $g$

If $\lim_{x \to \infty} \frac{f(x)}{g(x)} = 0$ , then $f$ grows slower than $g$

If $\lim_{x \to \infty} \frac{f(x)}{g(x)}$ is a positive number, they grow at the same rate

For example:

$\lim_{x \to \infty} \frac{x^3}{e^x} = 0$ , so $e^x$ grows faster than $x^3$

$\lim_{x \to \infty} \frac{\ln x}{\sqrt{x}} = 0$ , so $\sqrt{x}$ grows faster than $\ln x$

$\lim_{x \to \infty} \frac{4x^2 + x}{x^2} = 4$ , so $4x^2 + x$ and $x^2$ grow at the same rate

The rules below for rational functions are this same comparison: the limit at infinity depends on whether the numerator or the denominator grows faster. (L’Hospital’s Rule in Unit 4 gives a way to prove limits like

\frac{x^3}{e^x} \to 0

Rules for Horizontal Asymptotes of Rational Functions

For rational functions

\frac{f(x)}{g(x)}

Rule 1: Denominator Grows Faster

If the denominator grows faster than the numerator:

\lim_{x \to \infty} \frac{f(x)}{g(x)} = 0

Horizontal asymptote:

y = 0

Intuition: a small number divided by a huge number approaches zero.

Rule 2: Same Growth Rate

If the numerator and denominator grow at the same rate:

\lim_{x \to \infty} \frac{f(x)}{g(x)} = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}

Intuition: the ratio of the leading terms sets the asymptote.

Rule 3: Numerator Grows Faster

If the numerator grows faster than the denominator:

\lim_{x \to \infty} \frac{f(x)}{g(x)} = \pm\infty

No horizontal asymptote (the function grows without bound).

Finding Horizontal Asymptotes: Examples

Example 1: Denominator Grows Faster

Find

\lim_{x \to \infty} \dfrac{5x}{x^2 + 3}

Dominant terms:

Numerator: $5x$ (degree 1)

Denominator: $x^2$ (degree 2)

The denominator (

x^2

) grows faster than the numerator (

x

), so by Rule 1:

\lim_{x \to \infty} \frac{5x}{x^2 + 3} = 0

Answer: horizontal asymptote at

y = 0

Example 2: Same Growth Rate

Find

\lim_{x \to \infty} \dfrac{3x + 7}{x - 2}

Dominant terms:

Numerator: $3x$ (degree 1)

Denominator: $x$ (degree 1)

Both are degree 1, so they have the same growth rate. By Rule 2, look only at the leading coefficients:

\lim_{x \to \infty} \frac{3x + 7}{x - 2} = \frac{3}{1} = 3

Answer: horizontal asymptote at

y = 3

Note: the constants

+7

and

-2

don’t matter for end behavior.

Example 3: Higher Degree Polynomials

Find

\lim_{x \to \infty} \dfrac{2x^3 - 5x^2 + 7x - 1}{6x^3 + 4x - 8}

Focus only on the highest degree terms:

\lim_{x \to \infty} \frac{2x^3 - 5x^2 + 7x - 1}{6x^3 + 4x - 8} = \lim_{x \to \infty} \frac{2x^3}{6x^3} = \frac{2}{6} = \frac{1}{3}

Answer: horizontal asymptote at

y = \frac{1}{3}

Example 4: Numerator Grows Faster

Find

\lim_{x \to \infty} \dfrac{x^3}{2x + 5}

Compare degrees:

Numerator: degree 3

Denominator: degree 1

The numerator grows faster, so as

x

gets large, the numerator becomes much bigger than the denominator.

\lim_{x \to \infty} \frac{x^3}{2x + 5} = \infty

Answer: no horizontal asymptote (the function grows without bound)

Square Roots and Radicals

With square roots, you have to be careful about the sign of the result.

Example 5: Square Root in the Numerator

Find

\lim_{x \to \infty} \dfrac{\sqrt{4x^2 + 1}}{3x}

Simplify the square root for large positive

x

\sqrt{4x^2 + 1} \approx \sqrt{4x^2} = 2|x| = 2x \quad \text{(for } x > 0\text{)}

Evaluate:

\lim_{x \to \infty} \frac{2x}{3x} = \frac{2}{3}

Answer:

y = \frac{2}{3}

Example 6: Negative Infinity with a Square Root

Find

\lim_{x \to -\infty} \dfrac{\sqrt{4x^2 + 1}}{3x}

Critical distinction: when

x < 0

\sqrt{4x^2 + 1} \approx \sqrt{4x^2} = 2|x| = 2(-x) = -2x

The absolute value makes this positive, but since

x

is negative, we write it as

-2x

(which is positive).

Evaluate:

\lim_{x \to -\infty} \frac{-2x}{3x} = \frac{-2}{3} = -\frac{2}{3}

Answer:

y = -\frac{2}{3}

Important: this function has TWO different horizontal asymptotes.

As $x \to \infty$ : $y = \frac{2}{3}$

As $x \to -\infty$ : $y = -\frac{2}{3}$

Example 7: Even Power Under a Radical

Find the horizontal asymptotes of

f(x) = \frac{\sqrt{4x^4}}{3x^2}

Simplify the radical:

\sqrt{4x^4} = \sqrt{4} \cdot \sqrt{x^4} = 2x^2

Note that

\sqrt{x^4} = x^2

, which is always positive because it’s an even power.

As $x \to \infty$ :

\lim_{x \to \infty} \frac{2x^2}{3x^2} = \frac{2}{3}

As $x \to -\infty$ : since

x^4

is always positive (even power),

\lim_{x \to -\infty} \frac{2x^2}{3x^2} = \frac{2}{3}

Answer: only one horizontal asymptote,

y = \frac{2}{3}

, in both directions.

Exponential Functions and Limits at Infinity

Example 8: Exponential in the Numerator

Find

\lim_{x \to \infty} \dfrac{5e^x}{x^{100}}

Exponential functions grow much faster than any polynomial, so

\lim_{x \to \infty} \frac{5e^x}{x^{100}} = \infty

Answer: no horizontal asymptote

Example 9: Exponential with a Negative Exponent

Find

\lim_{x \to \infty} 3e^{-x}

Rewrite with a positive exponent:

3e^{-x} = \frac{3}{e^x}

x \to \infty

e^x \to \infty

, so

\lim_{x \to \infty} \frac{3}{e^x} = 0

Answer:

y = 0

Example 10: Exponential with a Fraction

Find

\lim_{x \to \infty} -4e^{\frac{1}{x}}

Step 1: Evaluate the exponent. As

x \to \infty

\frac{1}{x} \to 0

Step 2: Evaluate

e

to that power:

e^0 = 1

Step 3: Multiply by the coefficient:

-4 \cdot 1 = -4

Answer:

y = -4

Trigonometric Functions and Limits at Infinity

Special Trig Limit (Review)

\lim_{x \to 0} \frac{\sin x}{x} = 1

This ONLY works when $x \to 0$ .

Example 11: Sine Over a Large Number

Find

\lim_{x \to \infty} \dfrac{\sin x}{x}

Common mistake: thinking this equals 1. It doesn’t. That result only applies when

x \to 0

Correct reasoning:

Numerator: $\sin x$ oscillates between $-1$ and $1$ (bounded)

Denominator: $x \to \infty$ (unbounded)

\lim_{x \to \infty} \frac{\sin x}{x} = \frac{\text{bounded}}{\text{very large}} = 0

Answer:

y = 0

Example 12: Cosine of a Small Value

Find

\lim_{x \to \infty} -3\cos\left(\dfrac{1}{x}\right)

Step 1: As

x \to \infty

\frac{1}{x} \to 0

Step 2: Evaluate cosine:

\cos(0) = 1

Step 3: Multiply:

-3 \cdot 1 = -3

Answer:

y = -3

Example 13: Unbounded Oscillation

Find

\lim_{x \to \infty} \sin x

The sine function oscillates between

-1

and

1

forever and never settles on a single value.

Answer: does not exist (DNE), because the function oscillates without approaching a specific value.

Example 14: Growing Oscillation

Find

\lim_{x \to \infty} (5x \cos x)

While

\cos x

oscillates between

-1

and

1

, the factor

5x

grows without bound. The product oscillates with increasing amplitude:

Sometimes near $+5x$ (when $\cos x \approx 1$ )

Sometimes near $-5x$ (when $\cos x \approx -1$ )

The oscillation just gets bigger and bigger.

Answer: does not exist (DNE), because it oscillates with unbounded amplitude.

Practice Problems

Problem 1

Find

\lim_{x \to \infty} \dfrac{7x^2 - 3}{2x^2 + 5x}

Answer: same degree (both

x^2

), so the limit is

\frac{7}{2}

Problem 2

Find

\lim_{x \to \infty} \dfrac{x + 1}{x^3}

Answer: the denominator grows faster, so the limit is

0

Problem 3

Find

\lim_{x \to \infty} \dfrac{\sqrt{9x^2 + x}}{x}

Answer:

\sqrt{9x^2} = 3x

for

x > 0

, so the result is

\frac{3x}{x} = 3

Problem 4

Find

\lim_{x \to -\infty} \dfrac{\sqrt{9x^2 + x}}{x}

Answer:

\sqrt{9x^2} = 3|x| = -3x

for

x < 0

, so the result is

\frac{-3x}{x} = -3

Problem 5

Find

\lim_{x \to \infty} \dfrac{\cos x}{x^2}

Answer: bounded numerator over unbounded denominator, so the limit is

0

Key Takeaways

Horizontal asymptotes describe end behavior as $x \to \pm\infty$

Graphs can cross horizontal asymptotes, unlike vertical ones

Growth rate hierarchy: exponential beats polynomial beats logarithmic

Compare growth with a limit: if $\lim_{x \to \infty} \frac{f(x)}{g(x)} = \infty$ , $f$ grows faster; if it’s $0$ , $g$ grows faster

For rational functions, compare the degrees of the numerator and denominator

Only the highest degree terms matter for end behavior

With square roots, $\sqrt{x^2} = |x|$ , which equals $x$ if $x > 0$ and $-x$ if $x < 0$

Even powers under radicals are always positive: $\sqrt{x^4} = x^2$

$\lim_{x \to 0} \frac{\sin x}{x} = 1$ only works for $x \to 0$ , not $x \to \infty$

$\frac{\sin x}{x} \to 0$ as $x \to \infty$ (bounded over unbounded)

Unbounded trig oscillations mean the limit does not exist

A function can have different horizontal asymptotes on the left and right

Exponentials with negative exponents approach zero: $e^{-x} \to 0$ as $x \to \infty$

Back Next