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2.4: The Limit Laws - Limits at Infinity (Lecture Notes)

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Sep 9, 2024
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- 2.3: The Limit Laws - Limits at Finite Numbers (Lecture Notes)
- 2.5: The Precise Definition of a Finite Limit at a Finite Number (Lecture Notes)

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Finite Limits at Infinity and Horizontal Asymptotes

Definition: Finite Limit at Infinity (Informal)

If the values of $f(x)$ become arbitrarily close to the finite value $L$ as $x$ becomes sufficiently large, we say the function $f$ has a finite limit at infinity and write $\lim_{x \to \infty}f(x)=L. \nonumber$ A similar definition holds for $x \to -\infty$ .

Definition: Horizontal Asymptote

If $\displaystyle \lim_{x \to \infty}f(x)=L$ or $\displaystyle \lim_{x \to −\infty}f(x)=L$ , we say the line $y=L$ is a horizontal asymptote of $f$ .

Revisiting the Limit Laws

Theorem: Limit Laws for Limits at Infinity

Let $f(x)$ and $g(x)$ be defined for all $x \gt a$ , where $a$ is a real number. Assume that $L$ and $M$ are real numbers such that $\displaystyle \lim_{x \to \infty}{f(x)} = L$ and $\displaystyle \lim_{x \to \infty}{g(x)} = M$ . Let $c$ be a constant. Then, each of the following statements holds:

Sum and Difference Laws for Limits: $\displaystyle \lim_{x \to \infty}{(f(x) \pm g(x))} = \lim_{x \to \infty}{f(x)} \pm \lim_{x \to \infty}{g(x)} = L \pm M \nonumber$
Constant Multiple Law for Limits: $\displaystyle \lim_{x \to \infty}{cf(x)} = c \cdot \lim_{x \to \infty}{f(x)} = c L \nonumber$
Product Law for Limits: $\displaystyle \lim_{x \to \infty}{(f(x) \cdot g(x))} = \lim_{x \to \infty}{f(x)} \cdot \lim_{x \to \infty}{g(x)} = L \cdot M \nonumber$
Quotient Law for Limits: $\displaystyle \lim_{x \to \infty}\frac{f(x)}{g(x)} = \frac{\displaystyle \lim_{x \to \infty}f(x)}{\displaystyle \lim_{x \to \infty}g(x)}=\frac{L}{M} \nonumber$ for $M \neq 0$ .
Power Law for Limits: $\displaystyle \lim_{x \to \infty}\big(f(x)\big)^n = \big(\lim_{x \to \infty}f(x)\big)^n = L^n \nonumber$ for every positive integer $n$ .
Root Law for Limits: $\displaystyle \lim_{x \to \infty}\sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to \infty} f(x)}=\sqrt[n]{L} \nonumber$ for all $L$ if $n$ is odd and for $L \geq 0$ if $n$ is even.

Each of these Limit Laws can be adjusted for $x \to -\infty$ as long as $f(x)$ and $g(x)$ are defined for all $x \lt a$ , where $a$ is a real number.

Lemma

$\displaystyle \lim_{x \to \infty}{x} = \infty, \nonumber$ $\displaystyle \lim_{x \to -\infty}{x} = -\infty, \nonumber$ and $\displaystyle \lim_{x \to \pm \infty}{\frac{1}{x}} = 0. \nonumber$

Theorem

If $p \gt 0$ is a rational number, then $\displaystyle \lim_{x \to \infty}{\frac{1}{x^p}} = 0. \nonumber$ If $p \gt 0$ is a rational number such that $x^p$ is defined for all $x$ , then $\displaystyle \lim_{x \to -\infty}{\frac{1}{x^p}} = 0. \nonumber$

Proof: Let $p$ be a rational number. The only constraint on using the Limit Laws is that the values of the limits must be finite. Therefore, $a$ is allowed to be $\pm \infty$ . Thus,
$\begin{array}{rclcl} \displaystyle \lim_{x \to \infty}{\frac{1}{x^p}} & = & \displaystyle \lim_{x \to \infty}{\left(\frac{1}{x}\right)^p} & & \\ & = & \left(\displaystyle \lim_{x \to \infty}{\frac{1}{x}} \right)^p & & (\text{a mixture of the Power Law and Root Law for Limits}) \\ & = & 0^p & & (\text{by Lemma }\PageIndex{1}) \\ & = & 0 & & \\ \end{array} \nonumber$
If we allow $x \to -\infty$ , then additional restrictions on $p$ must be made. Specifically, since $p$ is a rational number, it cannot be equivalent to a simplified fraction with an even denominator. Otherwise, the even denominator would imply an even-indexed root of $x$ , and this would result in imaginary numbers. Hence, as long as $x^p$ is defined,
$\displaystyle \lim_{x \to -\infty}{\frac{1}{x^p}} = 0 \nonumber$
by a similar derivation.

Evaluating Finite Limits at Infinity

Lecture Example $\PageIndex{2}$ : Evaluating Finite Limits at Infinity Having Indeterminate Forms

Evaluate each of the following limits.

$\displaystyle \lim_{x \to \infty}{\frac{8.4 x^{15} - \pi x^9 + e}{4.2 x^{15} - x + 1.228}}$
$\displaystyle \lim_{x \to -\infty}{\frac{\sqrt{11 + 2x + 4x^2}}{5 x - 8}}$
$\displaystyle \lim_{x \to \infty}{\left(\sqrt{1 + x + 16x^2} - 4x\right)}$

Lecture Example $\PageIndex{3}$ : Evaluating Finite Limits at Infinity Graphically

Determine the horizontal asymptote(s) for the function.

$f(x) = e^{-x}$
$g(t) = \tanh{(t)}$

Theorem

$\lim_{x \to \infty}{\tan^{−1}{(x)}} = \frac{\pi}{2}, \nonumber$

$\lim_{x \to -\infty}{\tan^{−1}{(x)}} = −\frac{\pi}{2}, \nonumber$

$\lim_{x \to \infty}{\tanh{(x)}} = 1, \nonumber$

$\lim_{x \to -\infty}{\tanh{(x)}} = -1, \text{ and }\nonumber$

$\lim_{x \to \infty}{b^{-x}} = 0, \text{ where }b \gt 1. \nonumber$

Revisiting the Squeeze Theorem

Theorem: The Squeeze Theorem for Limits at Infinity

Let $f(x),g(x)$ , and $h(x)$ be defined for all $x \gt a$ , where $a$ is a real number. If

$f(x) \leq g(x) \leq h(x) \nonumber$

for all $x \gt a$ and

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

where $L$ is a real number, then $\displaystyle \lim_{x \to \infty}g(x)=L.$

Likewise, let $f(x),g(x)$ , and $h(x)$ be defined for all $x \lt a$ , where $a$ is a real number. If

$f(x) \leq g(x) \leq h(x) \nonumber$

for all $x \lt a$ and

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

where $L$ is a real number, then $\displaystyle \lim_{x \to -\infty}g(x)=L.$

Lecture Example $\PageIndex{4}$

Evaluate the limit.

$\displaystyle \lim_{x \to \infty}{ e^{-3x} \sin{(x)} }$
$\displaystyle \lim_{x \to \infty}{ e^{-3x} + \sin{(x)} }$
$\displaystyle \lim_{x \to \infty}{\dfrac{\lfloor x \rfloor}{x}}$

Infinite Limits at Infinity

Definition: Infinite Limit at Infinity (Informal)

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

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

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

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

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

Theorem

$\displaystyle \lim_{x \to \infty} x = \infty$
$\displaystyle \lim_{x \to -\infty} x = -\infty$
If $p \gt 0$ , $\displaystyle \lim_{x \to \infty} x^p = \infty$
If $b > 1$ , $\displaystyle \lim_{x \to \infty} b^x = \infty.$

Lecture Example $\PageIndex{5}$ : Evaluating an Infinite Limit at Infinity

Evaluate the limit.

$\lim_{x \to -\infty}{ \dfrac{2x + x^2 - x^3 + x^8}{x^5 + 8x^4 - x} }$
$\lim_{x \to \infty}{ e^x }$
$\lim_{x \to -\infty}{ (a x^6 + b x^{11}) },$ where $a, b \gt 0$ .

Finite Limits at Infinity and Horizontal Asymptotes

Definition: Finite Limit at Infinity (Informal)

Definition: Horizontal Asymptote

Revisiting the Limit Laws

Theorem: Limit Laws for Limits at Infinity

Lemma

Theorem

Evaluating Finite Limits at Infinity

Lecture Example 2.4.2\PageIndex{2}: Evaluating Finite Limits at Infinity Having Indeterminate Forms

Lecture Example 2.4.3\PageIndex{3}: Evaluating Finite Limits at Infinity Graphically

Theorem

Revisiting the Squeeze Theorem

Theorem: The Squeeze Theorem for Limits at Infinity

Lecture Example 2.4.4\PageIndex{4}

Infinite Limits at Infinity

Definition: Infinite Limit at Infinity (Informal)

Theorem

Lecture Example 2.4.5\PageIndex{5}: Evaluating an Infinite Limit at Infinity

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Lecture Example $\PageIndex{2}$ : Evaluating Finite Limits at Infinity Having Indeterminate Forms

Lecture Example $\PageIndex{3}$ : Evaluating Finite Limits at Infinity Graphically

Lecture Example $\PageIndex{4}$

Lecture Example $\PageIndex{5}$ : Evaluating an Infinite Limit at Infinity