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2.4: The Limit Laws - Limits at Infinity (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 writelimxf(x)=L.A similar definition holds for x.

Definition: Horizontal Asymptote

If limxf(x)=L or limxf(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>a, where a is a real number. Assume that L and M are real numbers such that limxf(x)=L and limxg(x)=M. Let c be a constant. Then, each of the following statements holds:

  • Sum and Difference Laws for Limits:limx(f(x)±g(x))=limxf(x)±limxg(x)=L±M
  • Constant Multiple Law for Limits:limxcf(x)=climxf(x)=cL
  • Product Law for Limits:limx(f(x)g(x))=limxf(x)limxg(x)=LM
  • Quotient Law for Limits:limxf(x)g(x)=limxf(x)limxg(x)=LMfor M0.
  • Power Law for Limits:limx(f(x))n=(limxf(x))n=Lnfor every positive integer n.
  • Root Law for Limits:limxnf(x)=nlimxf(x)=nLfor all L if n is odd and for L0 if n is even.

Each of these Limit Laws can be adjusted for x as long as f(x) and g(x) are defined for all x<a, where a is a real number.

Lemma

limxx=,limxx=,andlimx±1x=0.

Theorem

If p>0 is a rational number, thenlimx1xp=0.If p>0 is a rational number such that xp is defined for all x, thenlimx1xp=0.

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 ±. Thus,
limx1xp=limx(1x)p=(limx1x)p(a mixture of the Power Law and Root Law for Limits)=0p(by Lemma 2.4.1)=0
If we allow x, 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 xp is defined,
limx1xp=0
by a similar derivation.

Evaluating Finite Limits at Infinity

Lecture Example 2.4.2: Evaluating Finite Limits at Infinity Having Indeterminate Forms

Evaluate each of the following limits.

  1. limx8.4x15πx9+e4.2x15x+1.228
  2. limx11+2x+4x25x8
  3. limx(1+x+16x24x)
Lecture Example 2.4.3: Evaluating Finite Limits at Infinity Graphically

Determine the horizontal asymptote(s) for the function.

  1. f(x)=ex
  2. g(t)=tanh(t)
Theorem

limxtan1(x)=π2,

limxtan1(x)=π2,

limxtanh(x)=1,

limxtanh(x)=1, and 

limxbx=0, where b>1.

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>a, where a is a real number. If

f(x)g(x)h(x)

for all x>a and

limxf(x)=L=limxh(x)

where L is a real number, then limxg(x)=L.

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

f(x)g(x)h(x)

for all x<a and

limxf(x)=L=limxh(x)

where L is a real number, then limxg(x)=L.

Lecture Example 2.4.4

Evaluate the limit.

  1. limxe3xsin(x)
  2. limxe3x+sin(x)
  3. limxxx

Infinite Limits at Infinity

Definition: Infinite Limit at Infinity (Informal)

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

limxf(x)=.

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

limxf(x)=.

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

Theorem
  • limxx=
  • limxx=
  • If p>0, limxxp=
  • If b>1, limxbx=.
Lecture Example 2.4.5: Evaluating an Infinite Limit at Infinity

Evaluate the limit.

  1. limx2x+x2x3+x8x5+8x4x
  2. limxex
  3. limx(ax6+bx11), where a,b>0.

This page titled 2.4: The Limit Laws - Limits at Infinity (Lecture Notes) is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Roy Simpson.

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