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Section B.1 Limits at infinity
📝
I know. You thought your Calculus 1 instructor was crazy for spending sooooooo long on limits, especially when you were convinced you would never, ever use them, and now here they are again!
In this section, we’ll focus our attention on limits at infinity. If you want a refresher, check out your Calculus 1 book (available online--just search for "APEX calculus").
Exercises Exercises
1.
Evaluate each of the following limits.
(a)
\(\ds \lim_{b\to \infty} \frac{1}{12}\)
(b)
\(\ds \lim_{b\to \infty} \frac{1}{s},\) where
\(s\) is a constant
(c)
\(\ds \lim_{b\to \infty} e^{2b}\)
(d)
\(\ds \lim_{b\to \infty} e^{0.1b}\)
(e)
\(\ds \lim_{b\to \infty} e^{-2b}\)
2. For what values of
\(a\) is the limit
\(\ds \lim_{b \to \infty}e^{ab}\) finite?
3.
Evaluate each of the following limits.
(a)
\(\ds \lim_{b\to \infty} \left[ -\frac{1}{-3}e^{-3b} + \frac{1}{-3} \right]\)
(b)
\(\ds \lim_{b\to \infty} \left[ -\frac{1}{s}e^{-sb} + \frac{1}{s} \right]\)
where
\(s\) is a constant and
\(s \gt 0\)
(c)
\(\ds \lim_{b\to \infty} \left[ -\frac{1}{s}e^{-sb} + \frac{1}{s} \right]\)
where
\(s\) is a constant and
\(s \lt 0\text{.}\)
(d)
\(\ds \lim_{b\to \infty} \left[ \frac{1}{s^2}e^{-sb} \right]\)
where
\(s\) is a constant and
\(s \gt 0\text{.}\)
(e)
\(\ds \lim_{b\to\infty} \left[\frac{1}{s-3}e^{(3-s)b} + \frac{1}{3-s}\right]\)
where
\(s\) is a constant and
\(s \gt 3\)
(f)
\(\ds \lim_{b\to \infty} \left[ \frac{1}{s+7}e^{(-7-s)b} + \frac{1}{-7-s} \right]\)
where
\(s\) is a constant and
\(s \gt -7\)
(g)
\(\ds \lim_{b\to \infty} \left[ \frac{1}{s-a}e^{(a-s)b} + \frac{1}{a-s} \right]\)
where
\(s\) is a constant and
\(s \gt a\text{.}\)
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