4.6 Limits at infinity and asymptotes (Page 3/14)

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Definition

(Informal) We say a function $f$ has an infinite limit at infinity and write

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

if $f (x)$ becomes arbitrarily large for $x$ sufficiently large. We say a function has a negative infinite limit at infinity and write

lim_{x \to \infty} f (x) = − \infty .

if $f (x) < 0$ and $| f (x) |$ becomes arbitrarily large for $x$ sufficiently large. Similarly, we can define infinite limits as $x \to − \infty .$

Formal definitions

Earlier, we used the terms arbitrarily close , arbitrarily large , and sufficiently large to define limits at infinity informally. Although these terms provide accurate descriptions of limits at infinity, they are not precise mathematically. Here are more formal definitions of limits at infinity. We then look at how to use these definitions to prove results involving limits at infinity.

Definition

(Formal) We say a function $f$ has a limit at infinity , if there exists a real number $L$ such that for all $ε > 0,$ there exists $N > 0$ such that

| f (x) - L | < ε

for all $x > N .$ In that case, we write

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

(see [link] ).

We say a function $f$ has a limit at negative infinity if there exists a real number $L$ such that for all $ε > 0,$ there exists $N < 0$ such that

| f (x) - L | < ε

for all $x < N .$ In that case, we write

lim_{x \to − \infty} f (x) = L .

The function f(x) is graphed, and it has a horizontal asymptote at L. L is marked on the y axis, as is L + ॉ and L – ॉ. On the x axis, N is marked as the value of x such that f(x) = L + ॉ. — For a function with a limit at infinity, for all $x > N,$ $| f (x) - L | < ε .$

Earlier in this section, we used graphical evidence in [link] and numerical evidence in [link] to conclude that $lim_{x \to \infty} (\frac{2 + 1}{x}) = 2 .$ Here we use the formal definition of limit at infinity to prove this result rigorously.

Use the formal definition of limit at infinity to prove that $lim_{x \to \infty} (\frac{2 + 1}{x}) = 2 .$

Let $ε > 0 .$ Let $N = \frac{1}{ε} .$ Therefore, for all $x > N,$ we have

| 2 + \frac{1}{x} - 2 | = | \frac{1}{x} | = \frac{1}{x} < \frac{1}{N} = ε

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Use the formal definition of limit at infinity to prove that $lim_{x \to \infty} (\frac{3 - 1}{x^{2}}) = 3 .$

Let $ε > 0 .$ Let $N = \frac{1}{\sqrt{ε}} .$ Therefore, for all $x > N,$ we have

$| 3 - \frac{1}{x^{2}} - 3 | = \frac{1}{x^{2}} < \frac{1}{N^{2}} = ε$

Therefore, $lim_{x \to \infty} (3 - 1 / x^{2}) = 3 .$

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We now turn our attention to a more precise definition for an infinite limit at infinity.

Definition

(Formal) We say a function $f$ has an infinite limit at infinity and write

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

if for all $M > 0,$ there exists an $N > 0$ such that

f (x) > M

for all $x > N$ (see [link] ).

We say a function has a negative infinite limit at infinity and write

lim_{x \to \infty} f (x) = − \infty

if for all $M < 0,$ there exists an $N > 0$ such that

f (x) < M

for all $x > N .$

Similarly we can define limits as $x \to − \infty .$

The function f(x) is graphed. It continues to increase rapidly after x = N, and f(N) = M. — For a function with an infinite limit at infinity, for all $x > N,$ $f (x) > M .$

Earlier, we used graphical evidence ( [link] ) and numerical evidence ( [link] ) to conclude that $lim_{x \to \infty} x^{3} = \infty .$ Here we use the formal definition of infinite limit at infinity to prove that result.

Use the formal definition of infinite limit at infinity to prove that $lim_{x \to \infty} x^{3} = \infty .$

Let $M > 0 .$ Let $N = \sqrt[3]{M} .$ Then, for all $x > N,$ we have

x^{3} > N^{3} = {(\sqrt[3]{M})}^{3} = M .

Therefore, $lim_{x \to \infty} x^{3} = \infty .$

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Use the formal definition of infinite limit at infinity to prove that $lim_{x \to \infty} 3 x^{2} = \infty .$

Let $M > 0 .$ Let $N = \sqrt{\frac{M}{3}} .$ Then, for all $x > N,$ we have

$3 x^{2} > 3 N^{2} = 3 {(\sqrt{\frac{M}{3}})}^{2} 2 = \frac{3 M}{3} = M$

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End behavior

The behavior of a function as $x \to ± \infty$ is called the function’s end behavior . At each of the function’s ends, the function could exhibit one of the following types of behavior:

The function $f (x)$ approaches a horizontal asymptote $y = L .$
The function $f (x) \to \infty$ or $f (x) \to − \infty .$
The function does not approach a finite limit, nor does it approach $\infty$ or $− \infty .$ In this case, the function may have some oscillatory behavior.

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Source: OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2

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