Limits at Infinity and Asymptotes: Learn It 2

Limits at Infinity Cont.

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. 

limits at infinity (formal)

We say a function [latex]f[/latex] has a limit at infinity, if there exists a real number [latex]L[/latex] such that for all [latex]\varepsilon >0[/latex], there exists [latex]N>0[/latex] such that

[latex]|f(x)-L|<\varepsilon[/latex]

for all [latex]x>N[/latex]. In that case, we write

[latex]\underset{x\to \infty }{\lim}f(x)=L[/latex]

We say a function [latex]f[/latex] has a limit at negative infinity if there exists a real number [latex]L[/latex] such that for all [latex]\varepsilon >0[/latex], there exists [latex]N<0[/latex] such that

[latex]|f(x)-L|<\varepsilon[/latex]

for all [latex]x

[latex]\underset{x\to −\infty }{\lim}f(x)=L[/latex]
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 + ॉ.
Figure 9. For a function with a limit at infinity, for all [latex]x>N[/latex], [latex]|f(x)-L|<\varepsilon [/latex].

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

Use the formal definition of limit at infinity to prove that [latex]\underset{x\to \infty }{\lim}\left(2+\frac{1}{x}\right)=2[/latex].

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

infinite limit at infinity (formal)

We say a function [latex]f[/latex] has an infinite limit at infinity and write

[latex]\underset{x\to \infty }{\lim}f(x)=\infty[/latex]

if for all [latex]M>0[/latex], there exists an [latex]N>0[/latex] such that

[latex]f(x)>M[/latex]

for all [latex]x>N[/latex].

 

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

[latex]\underset{x\to \infty }{\lim}f(x)=−\infty[/latex]

if for all [latex]M<0[/latex], there exists an [latex]N>0[/latex] such that

[latex]f(x)

for all [latex]x>N[/latex].

Similarly we can define limits as [latex]x\to −\infty[/latex].

The function f(x) is graphed. It continues to increase rapidly after x = N, and f(N) = M.
Figure 10. For a function with an infinite limit at infinity, for all [latex]x>N[/latex], [latex]f(x)>M[/latex].

Earlier, we used graphical evidence and numerical evidence to conclude that [latex]\underset{x\to \infty }{\lim}x^3=\infty[/latex]. 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 [latex]\underset{x\to \infty }{\lim}x^3=\infty[/latex].