Section Limits involving Infinity and Asymptotes
In our original discussion of \(lim_{x\to a} f(x) = L\) we required that both \(a\) and \(L\) be real numbers. We now wish to consider the possibilities where either \(a= \pm \infty\) or \(L= \pm \infty\text{.}\) To determine \(lim_{x \to \infty} f(x)\) is to ask what is happening to the values of \(f(x)\) as \(x\) becomes large and positive. One possibility is that \(f(x)\) tends toward a real number as \(x\) tends toward infinity. In this case we write \(lim_{x\to \infty} f(x) = L\) and \(y=L\) is a horizontal asymptote. Another possibility is that \(f(x)\) grows without bound as \(x\) tends to infinity. In this case we write \(lim_{x \to \infty} f(x) = DNE(\infty)\) to indicate that the limit does not exist but \(f(x)\) tends to infinity. A limit does not exist unless it is a real number and infinity is not a number.
Now let's return to the case where \(a\) is a real number. It is possible that \(f(x)\) grows without bound as \(x\) approaches \(a.\) In this case we write \(lim_{x \to a} f(x) = DNE(\infty)\text{.}\) Now we know that the limit does not exist and \(x=a\) is a vertical asymptote.
Our definition for asymptote will be intuitive, like our definition for limits, because we don't give a mathematically valid explanation of what the word “approaches” means. Still, it will serve our purposes. A more precise definition might appear in a course titled “Real Analysis.”
Definition 3.8.
An asymptote of a function \(f\) is a (or line) that the graph of \(f\) approaches. A horizontal asymptote of \(f\) is a horizontal line that the graph of \(f\) approaches. A vertical asymptote of \(f\) is a vertical line that the graph of \(f\) approaches. A slant asymptote is a line that the graph approaches which is neither vertical nor horizontal.
Problem 3.9.
Graph \(\dsp f(x)=\frac{3x}{x-2}\text{.}\) What number does \(f(x)\) approach as \(x\) becomes large and positive? As \(x\) becomes large and negative? What happens to the values of \(f(x)\) as \(x\) approaches 2 from the left? From the right?
Definition 3.10.
Intuitive Definition We say that \(lim_{x\to +\infty} f(x)=L\) provided that as \(x\) becomes arbitrarily large and positive, the values of \(f(x)\) become arbitrarily closer to \(L\text{.}\)
Definition 3.11.
Another Intuitive Definition We say that \(lim_{x\to \infty} f(x)=\infty\) provided that as \(x\) becomes arbitrarily large and positive the values of \(f(x)\) become arbitrarily large and positive.
For the problems in this section, you may justify your answer using a graph or some algebra or limit tables or simply a sentence defending your answer, such as “the denominator gets larger and larger and the numerator is constant, so the limit must be zero.”
Problem 3.12.
Evaluate each of the following limits.
\(\dsp \lim_{x\to \infty} \frac{2x}{5x+3}\)
\(\dsp \lim_{x\to \infty } x^{2}+5\)
\(\dsp \lim_{t\to \infty } \frac{4t+cos(t)}{t}\)
Problem 3.13.
Evaluate each of the following limits or state why they do not exist.
\(\dsp \lim_{t\to -\infty } 2t^{3}+6\)
\(\dsp \lim_{z\to \infty } \frac{\sin(z)+7z}{1-z^{2}}\)
\(\dsp \lim_{\alpha \to -\infty } 3\sin(\alpha )\)
Here are precise restatements of our intuitive definitions.
Definition 3.14.
Precise Restatement of Definition 3.10 Let \(L\) be a real number. We say that \(lim_{x\to \infty} f(x)=L\) provided for each positive real number \(\epsilon\) there is a positive real number \(M\) such that for every \(x>M\text{,}\) we have \(|f(x)-L|\lt \epsilon\text{.}\)
Definition 3.15.
Precise Restatement of Definition 3.11 We say that \(lim_{x\to \infty} f(x)=\infty\) provided for each positive real number \(N\) there is a positive real number \(M\) such that for every \(x>M\text{,}\) we have \(f(x)>N\text{.}\)
Problem 3.16.
Based on the definitions above, write an intuitive definition and a precise definition for \(lim_{x\to \infty} f(x)=-\infty\)
Problem 3.17.
Suppose \(k\) is some real number and \(P\) be the second-degree polynomial \(P(x) = kx^2\text{.}\) Does the \(\lim_{x \to \infty} P(x)\) depend on the value of \(k\text{?}\) What is \(\lim_{x\to \infty } P(x)\text{?}\) What is \(\lim_{x\to -\infty } P(x)\text{?}\) Suppose \(C(x)=kx^3\text{.}\) What is \(\lim_{x\to \infty } C(x)\text{?}\) What is \(\lim_{x\to -\infty } C(x)\text{?}\)
Definition 3.18.
Yet Another Intuitive Definition We say that \(lim_{x\to a} f(x)=\infty\) if when \(x\) approaches \(a,\) the values of \(f(x)\) become arbitrarily large.
The next definition makes this idea precise.
Definition 3.19.
Precise Restatement of Definition 3.18 Let \(a\) be a real number. We say that \(lim_{x\to a} f(x)=\infty\) provided for each positive real number \(N\) there is positive number \(\epsilon\) so that for all \(x\) satisfying \(|x-a|\lt \epsilon\text{,}\) we have \(f(x)>M\)
Problem 3.20.
Evaluate each of the following limits. Graph each function; include any vertical asymptotes.
\(\dsp \lim_{t\to -2} \frac{3}{(2+t)^{2}}\)
\(\dsp \lim_{x\to 1^{+} } \frac{x^{2}-5x+6}{x^{3}-1}\)
Problem 3.21.
Evaluate each of the following limits.
\(\dsp \lim_{x\to 0^{-}} \frac{1}{x}\) and \(\dsp \lim_{x\to 0^{+}} \frac{1}{x}\)
\(\dsp \lim_{x\to -4^{-} } \frac{x-1}{x+4}\) and \(\dsp \lim_{x\to -4^{+} } \frac{x-1}{x+4}\)
\(\dsp \lim_{x\to 0 }\sin(\frac{1}{x})\)
Problem 3.22.
Find the asymptotes of each function and use them to sketch a rough graph of each.
\(\dsp G(x)=\frac{x-1}{x+4}\)
\(\dsp F(x)=\frac{3x}{5x-1}\)
\(\dsp g(x)=e^{-x}-2\)
\(\dsp h(z)=-\frac{2z+\sin(z)}{z}\)
Problem 3.23.
Write a statement that explains the relationship between limits and vertical asymptotes. Write a statement that explains the relationship between limits and horizontal asymptotes.