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

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

\[\lim_{x \to \infty}f(x)=L. \nonumber \]

If the values of \(f(x)\) becomes arbitrarily close to the finite value \(L\) for \(x<0\) as \(|x|\) becomes sufficiently large, we say that the function \(f\) has a finite limit at negative infinity and write

\[\lim_{x \to −\infty}f(x)=L. \nonumber \]

If the values \(f(x)\) are getting arbitrarily close to some finite value \(L\) as \(x \to \infty\) or \(x \to −\infty\), the graph of \(f\) approaches the line \(y=L\). In that case, the line \(y=L\) is a horizontal asymptote of \(f\).

Lecture Example \(\PageIndex{1}\)

Build a table of values to investigate the limit.

\[ \displaystyle \lim_{t \to -\infty}{ e^t \sin{(t)} } \nonumber \]

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 \(\PageIndex{1}\): 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 \(\PageIndex{1}\)

\[ \displaystyle \lim_{x \to \infty}{x} = \infty, \nonumber \]

\[ \displaystyle \lim_{x \to -\infty}{x} = -\infty, \nonumber \]

and

\[ \displaystyle \lim_{x \to \pm \infty}{\frac{1}{x}} = 0. \nonumber \]

Theorem \(\PageIndex{2}\)

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.

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

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

Determine the horizontal asymptote(s) for the function.

1. \( f(x) = e^{-x} \)
2. \( g(t) = \tanh{(t)} \)
3. \( h(x) = 2 - 2^{1/x} \)

Theorem \(\PageIndex{3}\)

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

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

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

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

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

Revisiting the Squeeze Theorem

Theorem \(\PageIndex{4}\): 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.

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

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 \(\PageIndex{5}\)

• \( \displaystyle \lim_{x \to \infty} x = \infty \)
• \( \displaystyle \lim_{x \to -\infty} x = -\infty \)
• 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.

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