\(0\) \(f(0)\) \(1\) \(f(0)\) is
undefined it is impossible to say
We introduce limits.
1 The basic idea
Consider the function
\[ f(x) = \frac {\sin (x)}{x}. \]
While \(f(x)\) is undefined at \(x=0\), we can still plot \(f(x)\) at other values near \(x = 0\).
![[Picture]](digInWhatIsALimit.online-0f3d0b323787cfb66a941585f8d57576.svg)
Use the graph of \(f(x) = \frac {\sin (x)}{x}\) above to answer the following question: What is \(f(0)\)?
Nevertheless, we can see that as \(x\) approaches zero, \(f(x)\) approaches one. From this setting we come to our definition of a limit.
Intuitively,
the limit of \(f(x)\) as \(x\) approaches \(a\) is \(L\),
written
\[ \lim _{x\to a} f(x) = L, \]
if the value of \(f(x)\) is as close as one wishes to \(L\) for all \(x\) sufficiently close, but not
equal to, \(a\). Use the graph of \(f(x) = \frac {\sin (x)}{x}\) above to finish the following statement: “A good guess is that…”
\(\lim _{x\to 0}\frac {\sin (x)}{x} = 1\). \(\lim _{x\to 1}\frac {\sin (x)}{x} = 0\). \(\lim _{x\to 1}f(x) = \frac {\sin (1)}{1}\). \(\lim _{x\to 0}f(x) = \frac {\sin (0)}{0}=\infty \).
Consider the following graph of \(y=f(x)\)
Use the graph to evaluate the following. Write DNE if the value does not
exist.
![[Picture]](digInWhatIsALimit.online-dd5fc7afbf65d79124aa52197ef86b03.svg)
- (a)
- \(f(-2) \begin{prompt}=\answer {1}\end{prompt}\)
- (b)
- \(\lim _{x\to -2}f(x)\begin{prompt}=\answer {1}\end{prompt}\)
- (c)
- \(f(-1) \begin{prompt}=\answer {2}\end{prompt}\)
- (d)
- \(\lim _{x\to -1}f(x) \begin{prompt}=\answer {2}\end{prompt}\)
- (e)
- \(f(0) \begin{prompt}=\answer {-2}\end{prompt}\)
- (f)
- \(\lim _{x\to 0} f(x) \begin{prompt}=\answer {0}\end{prompt}\)
- (g)
- \(f(1) \begin{prompt}=\answer {DNE}\end{prompt}\)
- (h)
- \(\lim _{x\to 1} f(x) \begin{prompt}=\answer {-2}\end{prompt}\)
2 Limits might not exist
Limits might not exist. Let’s see how this happens.
Consider the graph of \(f(x) = \lfloor x\rfloor \). ![[Picture]](digInWhatIsALimit.online-ebb9d1b58f0fb7142c5674fb5c2a8dcd.svg)
Explain why the limit
\[ \lim _{x\to 2} f(x) \]
does not exist. The function \(\lfloor x \rfloor \) is the function that
returns the greatest integer less than or equal to \(x\). Recall that ![[Picture]](digInWhatIsALimit.online-679b52576324f49670a64b6f5e3025c6.svg)
If we look at what happens for \(x\) values getting closer and closer to \(x=2\), but
staying on the left-side of 2, we see that \(f(x) = 1\). However, if we allow the values
of \(x\) getting closer and closer to \(x=2\), but staying on the right-side of 2, we see
![[Picture]](digInWhatIsALimit.online-d91900124a553adb0a9b510492d8193b.svg)
That is, for values of \(x\) close to \(2\) but larger than \(2\), \(f(x)=2\). We cannot find a single
number that \(f(x)\) approaches as \(x\) approaches 2, and so the limit does not exists.
\[ \lim _{x\to 2} \lfloor x \rfloor = L \]
if \(\lfloor x\rfloor \) can be
made arbitrarily close to \(L\) by making \(x\) sufficiently close, but not equal to, \(2\).
So let’s examine \(x\) near, but not equal to, \(2\). Now the question is: What is
\(L\)?
If this limit exists, then we should be able to look sufficiently close, but not at, \(x=2\), and see that \(f\) is approaching some number. Let’s look at a graph:
Tables can be used to help make guesses about limit values, but one must be careful.
Consider \(f(x) = \sin \left (\frac {\pi }{x}\right )\). Fill in the tables below rounding to three decimal places:
\[ \begin{array}{l|l} x & f(x) \\ \hline 0.1 & \begin{prompt}\answer {0}\end{prompt}\\ 0.01 & \begin{prompt}\answer {0}\end{prompt}\\ 0.001 & \begin{prompt}\answer {0}\end{prompt}\\ 0.0001 & \begin{prompt}\answer {0}\end{prompt}\\ \end{array} \]
We may
rush and say that, based on the table above, \[ \lim _{x\to 0}\sin \left (\frac {\pi }{x}\right )=0. \]
But, recall the definition of the limit: \(L\)
is the limit if the value of \(f(x)\) is as close as one wishes to \(L\) for all \(x\) sufficiently close, but
not equal to, \(a\).
From this table we can see that \(f(x)(=0)\) is as close as one wishes to \(L(=0)\) for some values \(x\) that are sufficiently close to \(a(=0)\). But this does does not satisfy the definition of the limit, at least, not yet.
But, wait! Fill in another table, rounding to three decimal places.
\[ \begin{array}{l|l} x & f(x) \\ \hline 0.3 & \begin{prompt}\answer {-.866}\end{prompt} \\ 0.03 & \begin{prompt}\answer {-.866}\end{prompt} \\ 0.003 & \begin{prompt}\answer {-.866}\end{prompt} \\ 0.0003 & \begin{prompt}\answer {-.866}\end{prompt} \end{array} \]
What do
these two tables tell us about \[ \lim _{x\to 0}\sin \left (\frac {\pi }{x}\right )? \]
\(\lim _{x\to 0}\sin \left (\frac {\pi }{x}\right ) = 0\) \(\lim _{x\to 0}\sin \left (\frac {\pi }{x}\right )=1\) \(\lim _{x\to 0}\sin \left (\frac {\pi }{x}\right ) = -.866\) \(\lim _{x\to 0}\sin \left (\frac {\pi }{x}\right ) = -.433\) The limit does not exist.
The
limit does not exist. The first table shows that we can always find a value
of \(x\) as close as we want to \(0\) such that \(f(x)=0\). However, the limit is not equal to \(0\),
since the second table shows that we can also find a value of \(x\) as close as we
want to \(0\) such that \(f(x)=-\frac {\sqrt {3}}{2}\). It turns out that for any number \(y\), \(-1\le y\le 1\), we can find a value
of \(x\) as close as we want to \(0\) such that \(f(x)=y\). Check the graph of the function \(f\).
![[Picture]](digInWhatIsALimit.online-75a92ed18638d7d0cd9dc9b83d89ce1e.svg)
We see that \(f(x)\)
oscillates “wildly” as \(x\) approaches \(0\), and hence does not approach any one number.
3 One-sided limits
While we have seen that \(\lim _{x\to 2}\lfloor x\rfloor \) does not exist, its graph looks much “nicer"near \(a=2\) than does the previous graph near \(a=0\). More can be said about the function \(\lfloor x\rfloor \) and its behavior near \(a=2\).
Intuitively,
for the function \(f\), \(L\) is the limit from the right as \(x\) approaches \(a\),
written
\[ \lim _{x\to a^+} f(x) = L, \]
if the value of \(f(x)\) is as close as one wishes to \(L\) for all \(x>a\) sufficiently close to
\(a\).
Similarly,
for the function \(f\), \(L\) is the limit from the left as \(x\) approaches \(a\),
written
\[ \lim _{x\to a^-} f(x) = L, \]
if the value of \(f(x)\) is as close as one wishes to \(L\) for all \(x<a\) sufficiently close to
\(a\).
Compute: ![[Picture]](digInWhatIsALimit.online-e663739dd3c5b3c4087b48ca320131ea.svg)
\[ \lim _{x\to 2^-} f(x)\qquad \text {and}\qquad \lim _{x\to 2^+} f(x) \]
by using the graph below From the graph we can see that as \(x\) approaches \(2\) from the left, \(\lfloor x\rfloor \) remains at \(y=1\) up until
the exact point that \(x=2\). Hence
\[ \lim _{x\to 2^-} f(x)=1. \]
Also from the graph we can see that as \(x\) approaches \(2\)
from the right, \(\lfloor x\rfloor \) remains at \(y=2\) up to \(x=2\). Hence \[ \lim _{x\to 2^+} f(x)=2. \]
4 When you put this all together
One-sided limits help us talk about limits.
A limit
\[ \lim _{x \to a} f(x) \]
exists if and only if
- \(\lim _{x \to a^-} f(x)\) exists
- \(\lim _{x \to a^+} f(x)\) exists
- \(\lim _{x \to a^-} f(x) = \lim _{x \to a^+} f(x)\)
In this case, \(\lim _{x \to a} f(x)\) is equal to the common value of the two one sided limits.
Evaluate the expressions by referencing the graph below. Write DNE if the limit
does not exist. ![[Picture]](digInWhatIsALimit.online-627b53da72131c62725973c0e48258c0.svg)
- (a)
- \(\lim _{x\to 4} f(x) \begin{prompt}=\answer {8}\end{prompt}\)
- (b)
- \(\lim _{x\to -3} f(x)\begin{prompt}=\answer {6}\end{prompt}\)
- (c)
- \(\lim _{x\to -3^-} f(x) \begin{prompt}=\answer {6}\end{prompt}\)
- (d)
- \(\lim _{x\to 0} f(x) \begin{prompt}=\answer {DNE}\end{prompt}\)
- (e)
- \(\lim _{x\to 0^-} f(x) \begin{prompt}=\answer {-2}\end{prompt}\)
- (f)
- \(\lim _{x\to 0^+} f(x) \begin{prompt}=\answer {-1}\end{prompt}\)
- (g)
- \(f(0) \begin{prompt}=\answer {-3/2}\end{prompt}\)
- (h)
- \(f(-2) \begin{prompt}=\answer {8}\end{prompt}\)
- (i)
- \(\lim _{x\to -2^+} f(x) \begin{prompt}=\answer {2}\end{prompt}\)
- (j)
- \(\lim _{x\to -2^-} f(x) \begin{prompt}=\answer {6}\end{prompt}\)
- (k)
- \(\lim _{x\to 2^-} f(x) \begin{prompt}=\answer {7}\end{prompt}\)
- (l)
- \(\lim _{x\to 1} f(x) \begin{prompt}=\answer {3}\end{prompt}\)