3.3: Limits

The notion of a limit is closely related to that of a derivative, but it is more general. In this chapter \(f\) will always be a real function of one variable. Let us recall the definition of the slope off at a point \(a\):

\(S\) is the slope of \(f\) at \(a\) if whenever \(dx\) is infinitely close to but not equal to zero, the quotient \[\frac{f(a + \Delta x) - f(a)}{\Delta x} \nonumber\]is infinitely close to S.

We now define the limit. \(c\) and \(L\) are real numbers.

Definition:

\(L\) is the limit of \(f(x)\) as \(x\) approaches \(c\) if whenever \(x\) is infinitely close to but not equal to \(c\), \(f(x)\) is infinitely close to \(L\).

In symbols, \[\lim_{x \rightarrow c} f(x) = L \nonumber\]if whenever \(x \approx c\) but \(x \neq c\), \(f(x) \approx L\). When there is no number \(L\) satisfying the above definition, we say that the limit of \(f(x)\) as \(x\) approaches \(c\) does not exist.

Notice that the limit \[\lim_{x \rightarrow c} f(x) = L \nonumber\]depends only on the values of \(f(x)\) for \(x\) infinitely close but not equal to \(c\). The value \(f(c)\) itself has no influence at all on the limit. In fact, it very often happens that \[\lim_{x \rightarrow c} f(x) = L \nonumber\]exists but \(f(c)\) is undefined.

Figure \(\PageIndex{1(a)}\) shows a typical limit. Looking at the point \((c, L)\) through an infinitesimal microscope, we can see the entire portion of the curve with \(x \approx c\) because \(f(x)\) will be infinitely close to \(L\) and hence within the field of vision of the microscope.

In Figure \(\PageIndex{1(b)}\), part of the curve with \(x \approx c\) is outside the field of vision of the microscope, and the limit does not exist.

Our first example of a limit is the slope of a function.

Figure \(\PageIndex{1}\): Points on a curve where limits exist and do not exist.

Theorem \(\PageIndex{1}\)

The slope of \(f\) at \(a\) is given by the limit \[f'(a) = \lim_{\Delta x \rightarrow 0} \frac{f(a + \Delta x) - f(a)}{\Delta x}. \nonumber\]

Verbally, the slope of \(f\) at \(a\) is the limit of the ratio of the change in \(f(x)\) to the change in \(x\) as the change in \(x\) approaches zero. The theorem is seen by simply
comparing the definitions of limit and slope. The slope exists exactly when the limit exists; and when they do exist they are equal. Notice that the ratio \[\frac{f(a + \Delta x) - f(a)}{\Delta x} \nonumber\] is undefined when \(\Delta x = 0\).

The slope of \(f\) at \(a\) is also equal to the limit \[f'(a) = \lim_{x \rightarrow a} \frac{f(x) - f(a)}{x - a}. \nonumber\]

This is seen by setting \[\begin{align*} \Delta x &= x - a, \\ x &= a + \Delta x. \end{align*}\]

Then when \(x \approx a\) but \(x \neq a\), we have \(\Delta x \approx 0\) but \(\Delta x \neq 0\); and \[\frac{f(x) - f(a)}{x - a} = \frac{f(a + \Delta x) - f(a)}{\Delta x} \approx f'(a). \nonumber\]

Sometimes a limit can be evaluated by recognizing it as a derivative and using Theorem \(\PageIndex{1}\) above.

Example \(\PageIndex{1}\)

Evaluate \[\lim_{\Delta x \rightarrow 0} \frac{(3 + \Delta x)^{2} - 9}{\Delta x}. \nonumber\]

Solution

Let \(F(x) = x^{2}\). The given limit is just \(F'(3)\), \[\begin{align*} F'(3) &= \lim_{\Delta x \rightarrow 0} \frac{F(3 + \Delta x)^{2} - 9}{\Delta x} = 6. \\ F'(3) = 2 \cdot 3 = 6. \end{align*}\]

Therefore \[\lim_{\Delta x \rightarrow 0} \frac{(3 + \Delta x)^{2} - 9}{\Delta x} = 6. \nonumber\]

The symbol \(x\) in \[\lim_{x \rightarrow c} f(x) \nonumber\]is an example of a "dummy variable." The value of the limit does not depend on \(x\) at all. However, it does depend on \(c\). If we replace \(c\) by a variable \(u\), we obtain a new function \[L(u) = \lim_{x \rightarrow u} f(x). \nonumber\]

A limit \(\lim_{x \rightarrow c} f(x)\) is usually computed as follows.

Step 1: Let \(x\) be infinitely close but not equal to \(c\), and simplify \(f(x)\).
Step 2: Compute the standard part \(st(f(x))\).

Conclusion: If the limit \(\lim_{x \rightarrow c} f(x)\) exists, it must equal \(st(f(x))\).

Example \(\PageIndex{1}\) (continued)

Instead of using the derivative, we can directly compute \[\lim_{\Delta x \rightarrow 0} \frac{(3 + \Delta x)^{2} - 9}{\Delta x}. \nonumber\]

Solution

Step 1
Let \(\Delta x \approx 0\), but \(\Delta x \neq 0\). Then \[\frac{(3 + \Delta x)^{2} - 9}{\Delta x} = \frac{(9 + 6 \Delta x + \Delta x^{2} - 9}{\Delta x} = \frac{6 \Delta x + \Delta x^{2}}{\Delta x} = 6 + \Delta x. \nonumber\]

Step 2
Taking standard parts, \[st \frac{(3 + \Delta x)^{2} - 9}{\Delta x} = st(6 + \Delta x) = 6. \nonumber\]

Therefore the limit is equal to 6. (See Figure \(\PageIndex{2}.)

Figure \(\PageIndex{2}\): Graph of \(f(\Delta x) = \dfrac{(3 + \Delta x)^{2} - 9}{\Delta x}\), showing the limit is 6 as \(\Delta x\) approaches 0.

Example \(\PageIndex{2}\)

Find \(\displaystyle \lim_{t \rightarrow 4} \left(t^{2} + 3t - 5\right)\).

Solution

Step 1
Let \(t\) be infinitely close to but not equal to \(4\).

Step 2
We take the standard part \[st \left(t^{2} + 3t - 5\right) = 4^{2} + 3 \cdot 4 - 5 = 23, \nonumber\] so the limit is 23. 

Example \(\PageIndex{3}\)

Find \(\displaystyle \lim_{x \rightarrow 2} = \dfrac{x^{2} + 3x - 10}{x^{2} - 4}\).

Solution

Step 1 
This time the term inside the limit is undefined at \(x = 2\). Taking \(x \approx 2\) but \(x \neq 2\), we have \[\frac{x^{2} + 3x - 10}{x^{2} - 4} = \frac{(x - 2)(x+5)}{(x - 2)(x + 2)} = \frac{x + 5}{x + 2}. \nonumber\]

Step 2
\[st \left(\frac{x^{2} + 3x - 10}{x^{2} - 4}\right) = st \left(\frac{x+5}{x+2}\right) = \frac{2 + 5}{2 + 2} = \frac{7}{4}. \nonumber\]

Thus \[\lim_{x \rightarrow 2} \left(\frac{x^{2} + 3x - 10}{x^{2} - 4}\right) = \frac{7}{4}. \nonumber\]

Example \(\PageIndex{4}\)

Find \(\displaystyle \lim_{x \rightarrow 0} \left( \dfrac{(2/x) + 3}{(3/x) - 1} \right)\).

Solution

Step 1
Taking \(x \approx 0\) but \(x \neq 0\), \[\frac{(2/x) + 3}{(3/x) - 1} = \frac{2 + 3x}{3 - x}. \nonumber\]

Step 2
\[st \left( \frac{(2/x) + 3}{(3/x) - 1} \right) = st \left(\frac{2 + 3x}{3 - x}\right) = \frac{2}{3}. \nonumber\]

Thus the limit exists and equals \(\frac{2}{3}\).

Example \(\PageIndex{5}\)

Find \(\displaystyle \lim_{x \rightarrow 9} \left( \dfrac{\sqrt{x} - 3}{x - 9} \right)\).

Solution

Step 1
Taking \(x \approx 9\) but \(x \neq 9\), \[\frac{\sqrt{x} - 3}{x - 9} = \frac{\left(\sqrt{x} - 3\right) \left(\sqrt{x} + 3\right)}{(x - 9) \left(\sqrt{x} + 3\right)} = \frac{x - 9}{(x - 9) \left(\sqrt{x} + 3\right)} = \frac{1}{\sqrt{x} + 3}. \nonumber\]

Step 2
\[st \left( \dfrac{\sqrt{x} - 3}{x - 9} \right) = st \left(\frac{1}{\sqrt{x} + 3}\right) = \frac{1}{\sqrt{9} + 3} = \frac{1}{6}, \nonumber\]so the limit exists and is \(\frac{1}{6}\).

Our rules for standard parts in Chapter 1 lead at once to rules for limits. We list these rules in Table \(\PageIndex{1}\). The limit rules can be applied whenever the two limits \(\displaystyle \lim_{x \rightarrow c} f(x)\) and \(\displaystyle \lim_{x \rightarrow c} g(x)\) exist.

Table \(\PageIndex{1}\). Rules for standard parts as applied to limits.

Standard Part Rule	Limit Rule
\(st(kb) = k \ st(b), \ k \text{ real}\)	\(\displaystyle \lim_{x \rightarrow c} kf(x) = k \lim_{x \rightarrow c} f(x)\)
\(st(a + b) = st(a) + st(b)\)	\(\displaystyle \lim_{x \rightarrow c} \left(f(x) + g(x)\right) = \lim_{x \rightarrow c} f(x) + \lim_{x \rightarrow c} g(x)\)
\(st(ab) = st(a) \cdot st(b)\)	\(\displaystyle \lim_{x \rightarrow c} \left(f(x) g(x)\right) = \lim_{x \rightarrow c} f(x) \cdot \lim_{x \rightarrow c} g(x)\)
\(st(a/b) = st(a) / st(b)\)	\(\displaystyle \lim_{x \rightarrow c} \left(f(x) / g(x)\right) = \lim_{x \rightarrow c} f(x) / \lim_{x \rightarrow c} g(x), \text{ if } \lim_{x \rightarrow c} g(x) \neq 0\)
\(st \left(\sqrt[n]{a}\right) = \sqrt[n]{st(a)}, \text{ if } a > 0\)	\(\displaystyle \lim_{x \rightarrow c} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \rightarrow c} f(x)}, \text{ if } \lim_{x \rightarrow c} f(x) > 0\)

Example \(\PageIndex{6}\)

Find \(\displaystyle \lim_{x \rightarrow 1} \left(x^{2} - 2x\right) \sqrt{\left(x^{2} - 1\right) / (x - 1)}.\)

Solution

All the limits involved exist, so we can use the limit rules to compute the limit as follows. First we find the limit of the expression inside the radical. 

\[\lim_{x \rightarrow 1} \frac{x^{2} - 1}{x - 1} = \lim_{x \rightarrow 1} \frac{(x - 1)(x + 1)}{x - 1} = \lim_{x \rightarrow 1} (x + 1) = 2. \nonumber\]

Now we find the answer to the original problem.

\[\begin{align*} \lim_{x \rightarrow 1} \left(x^{2} - 2x\right) \sqrt{\left(x^{2} - 1\right) / (x - 1)} &= \lim_{x \rightarrow 1} \left(x^{2} - 2x\right) \sqrt{\lim_{x \rightarrow 1} \left(x^{2} - 1\right) / (x - 1)} \\ &= (1 - 2) \sqrt{2} = - \sqrt{2}. \end{align*}\]

There are three ways in which a limit \(\displaystyle \lim_{x \rightarrow c} f(x)\) can fail to exist:

1. \(f(x)\) is undefined for some \(x\) which is infinitely close but not equal to \(c\).
2. \(f(x)\) is infinite for some \(x\) which is infinitely close but not equal to \(c\).
3. The standard part of \(f(x)\) is different for different numbers \(x\) which are infinitely close but not equal to \(c\).

Example \(\PageIndex{7}\)

\(\displaystyle \lim_{x \rightarrow 0} \sqrt{x}\) does not exist because \(\sqrt{x}\) is undefined for negative infinitesimal \(x \neq 0\). (See Figure \(\PageIndex{3(a)}\).)

Example \(\PageIndex{8}\)

\(\displaystyle \lim_{x \rightarrow 0} 1/x^{2}\) does not exist because \(1/x^{2}\) is infinite for infinitesimal \(x \neq 0\). (See Figure \(\PageIndex{3(b)}\).)

Figure \(\PageIndex{3}\): Graphs of (a) \(y = \sqrt{x}\), (b) \(y = \dfrac{1}{x^{2}}\), and (c) \(y = \dfrac{x}{|x|}\).

Example \(\PageIndex{9}\)

\(\displaystyle \lim_{x \rightarrow 0} x/|x|\) does not exist because \[st \left(\frac{x}{|x|}\right) = \begin{cases} 1 \quad &\text{if } x > 0, \\ -1 &\text{if } x < 0. \end{cases} \nonumber\]

(See Figure \(\PageIndex{3(c)}\)).

In the above examples the function behaves differently on one side of the point 0 than it does on the other side. For such functions, one-sided limits are useful.

We say that \[\lim_{x \rightarrow c^{+}} f(x) = L \nonumber\]if whenever \(x > c\) and \(x \approx c\), \(f(x) \approx L\).

\[\lim_{x \rightarrow c^{-}} f(x) = L \nonumber\]means that whenever \(x < c\) and \(x \approx c\), \(f(x) \approx L\). These two kinds of limits, shown in Figure \(\PageIndex{4}\), are called the limit from the right and the limit from the left.

Figure \(\PageIndex{4}\): One-sided limits.

Theorem \(\PageIndex{2}\)

A limit has value \(L\), \[\lim_{x \rightarrow c} f(c) = L, \nonumber\]

if and only if both one-sided limits exist and are equal to \(L\), \[\lim_{x \rightarrow c^{-}} f(c) = \lim_{x \rightarrow c^{+}} f(c) = L. \nonumber\]

Proof

If \(\displaystyle \lim_{x \rightarrow c\) f(x) = L\), it follows at once from the definition that both one-sided limits are \(L\).

Assume that both one-sided limits are equal to \(L\). Let \(x \approx c\), but \(x \neq c\). Then either \(x < c\) or \(x > c\). If \(x < c\), then because \(\lim_{x \rightarrow c^{-}} f(x) = L\), we have \(f(x) \approx L\). On the other hand if \(x > c\), then \(\lim_{x \rightarrow c^{+}} f(x) = L\) gives \(f(x) \approx L\). Thus in either case \(f(x) \approx L\). This shows that \(\lim_{x \rightarrow c} f(x) = L\).

When a limit does not exist, it is possible that neither one-sided limit exists, that just one of them exists, or that both one-sided limits exist but have different values.

Example \(\PageIndex{7}\) (continued)

\(\displaystyle \lim_{x \rightarrow 0^{+}} \sqrt{x} = 0\), and \(\displaystyle \lim_{x \rightarrow 0^{-}} \sqrt{x}\) does not exist.

Example \(\PageIndex{8}\) (continued)

Neither \(\displaystyle \lim_{x \rightarrow 0^{+}} 1/x^{2}\) nor \(\displaystyle \lim_{x \rightarrow 0^{-}} 1/x^{2}\) exists.

Example \(\PageIndex{9}\) (continued)

\(\displaystyle \lim_{x \rightarrow 0^{+}} x/|x| = 1\), and \(\displaystyle \lim_{x \rightarrow 0^{-}} x/|x| = -1\). 

Problems for Section 3.3

In each problem below, determine whether or not the limit exists. When the limit exists, find its value. With a calculator, compute some values as \(x\) approaches its limit, and see what happens.

1.	\(\displaystyle \lim_{t \rightarrow 4} 3t^{2} + t + 1\)	2.	\(\displaystyle \lim_{\Delta x \rightarrow -1} \dfrac{(\Delta x)^{2} + 2 \Delta x + 1}{\Delta x + 1}\)
3.	\(\displaystyle \lim_{x \rightarrow c} \sqrt{c - x}\)	4.	\(\displaystyle \lim_{y \rightarrow 0} \dfrac{1}{y^{5}}\)
5.	\(\displaystyle \lim_{x \rightarrow 2} \dfrac{x}{x^{2} - 4}\)	6.	\(\displaystyle \lim_{x \rightarrow 2} \dfrac{x^{2} - 4}{x - 2}\)
7.	\(\displaystyle \lim_{v \rightarrow 8} \dfrac{\sqrt{8} - \sqrt{v}}{v - 8}\)	8.	\(\displaystyle \lim_{x \rightarrow 0} \dfrac{\sqrt{x + 1} - 1}{x}\)
9.	\(\displaystyle \lim_{u \rightarrow 1} \dfrac{\sqrt[3]{u} - 1}{u - 1}\)	10.	\(\displaystyle \lim_{t \rightarrow 0} \dfrac{t^{3} - 2t^{2} + 4}{3t^{2} - 5t + 7}\)
11.	\(\displaystyle \lim_{y \rightarrow 0} \left(\sqrt{1 + 1/y} - \sqrt{1/y}\right)\)	12.	\(\displaystyle \lim_{x \rightarrow 0} \dfrac{(a + x)^{2} - a^{2}}{x}\)
13.	\(\displaystyle \lim_{y \rightarrow -1} \dfrac{y^{2} + 1}{y + 1}\)	14.	\(\displaystyle \lim_{x \rightarrow 1} \dfrac{|x - 1|}{x - 1}\)
15.	\(\displaystyle \lim_{x \rightarrow 1^{+}} \dfrac{|x - 1|}{x - 1}\)	16.	\(\displaystyle \lim_{x \rightarrow c^{-}} \sqrt{c - x}\)
17.	\(\displaystyle \lim_{z \rightarrow 1} \sqrt{z + \sqrt{z + \sqrt{z}}}\)	18.	\(\displaystyle \lim_{x \rightarrow a} \sqrt{|a - x|}\)
19.	\(\displaystyle \lim_{x \rightarrow 0^{+}} x \sqrt{1 + x^{-2}}\)	20.	\(\displaystyle \lim_{x \rightarrow 0^{-}} x \sqrt{1 + x^{-2}}\)
21.	\(\displaystyle \lim_{t \rightarrow 0} \dfrac{1 + 2t^{-1}}{3 - 4t^{-1}}\)	22.	\(\displaystyle \lim_{x \rightarrow 0} \dfrac{3 + 4x^{-1} - 5x^{-2}}{6 - x^{-1} + 3x^{-2}}\)
23.	\(\displaystyle \lim_{\Delta x \rightarrow 0} \dfrac{(x + \Delta x)^{2} - x^{2}}{\Delta x}\)	24.	\(\displaystyle \lim_{\Delta x \rightarrow 0} \dfrac{\frac{1}{x + \Delta x} - \frac{1}{x}}{\Delta x} \ (x \neq 0)\)
25.	\(\displaystyle \lim_{\Delta t \rightarrow 0} \dfrac{\sqrt{t + \Delta t} - \sqrt{t}}{\Delta t} \ (t > 0)\)	26.	\(\displaystyle \lim_{\Delta t \rightarrow 0} \dfrac{(t + \Delta t)^{1/5} - t^{1/5}}{\Delta t} \ (t > 0)\)
27.	\(\displaystyle \lim_{\Delta x \rightarrow 0} \dfrac{(x - \Delta x)^{3} - x^{3}}{\Delta x}\)	28.	\(\displaystyle \lim_{\Delta x \rightarrow 0} \dfrac{\frac{x + \Delta x}{x + \Delta x + 1} - \frac{x}{x + 1}}{\Delta x}\)
29.	\(\displaystyle \lim_{\Delta x \rightarrow 0^{-}} \dfrac{|(1 + \Delta x)^{3} - (1 + \Delta x)|}{\Delta x}\)	30.	\(\displaystyle \lim_{\Delta x \rightarrow 0^{+}} \dfrac{|(1 + \Delta x)^{3} - (1 + \Delta x)|}{\Delta x}\)
31.	\(\displaystyle \lim_{\Delta x \rightarrow 0^{-}} \dfrac{\sqrt{1 - (1 + \Delta x)^{2}}}{\Delta x}\)