Fact 2.3.1. The Constant Factor Rule for Limits.
\begin{equation*}
\lim_{x \to a} [C \cdot f(x)] = C \lim_{x \to a}f(x).
\end{equation*}
Section 2.3 The Limit Laws
References.
- Calculus, Early Transcendentals by Stewart, Section 2.3.
We now have the fundamental idea of limits, and are ready to learn how to compute limits \(\lim_{x \to a} f(x)\) as easily as possible, and to use this to compute the tangent slopes of curves, instantaneous velocities and such as easily as possible. Building this collection of calculational skills is the main goal of the rest of this chapter, with some related ideas and applications mixed in.
Limits for Two Very Basic Functions.
We start with two very simple and intuitive cases, which are then surprisingly useful in handling other functions.
Firstly, the limit of a constant function \(f(x)=c\) is easy:
for any value of \(x\text{,}\) \(f(x)\) is extremely close to (in fact equal to!) \(c\) and so as \(x \to a\) the limit is \(c\text{.}\) That is,
\begin{equation*}
\lim_{x \to a} c = c.
\end{equation*}
Next, almost as easy, is to note that for \(f(x)=x\text{,}\) as \(x\) approaches \(a\text{,}\) \(f(x)=x\) also approaches \(a\) since it is the same quantity. That is,
\begin{equation*}
\lim_{x \to a} x = a.
\end{equation*}
Note: In each case, the limit is just the value of \(f(a)\text{,}\) which is true for many "nice" functions, but not for all functions: we have already seen some exceptions above. This is our first sighting of continuity, which we will explore more in Section 2.4.
These two results are the building blocks that allow us to easily compute limits for any polynomial or rational function, once we know how to put together information about the limits of simple functions to get limits of more complicated ones.
The Limit of a Constant Multiple of a Function.
The first combining rule comes from this intuitive idea: when \(f(x)\) is close to L, \(C \cdot f(x)\) is close to \(C \cdot L\text{.}\)
Thus if \(f(x) \to L\) as \(x \to a\text{,}\) then \(C \cdot f(x) \to C \cdot L\) as \(x \to a\text{.}\)
The first half of this says that \(\displaystyle\lim_{x \to a}f(x) = L\text{,}\) so the second half gives
Example 2.3.2.
Combining this with the previous result
\begin{equation*}
\lim_{x \to 2} 7x
= 7 \lim_{x \to 2} x
= 7 \cdot 2
= 14.
\end{equation*}
The Limits of Sums and Differences of Functions.
The next basic idea is that:
when \(f(x) \) is close to \(L \) and \(g(x) \) is close to \(M \text{,}\) their sum \(f(x)+g(x) \) is close to \(L+M \text{.}\)
This leads to
Fact 2.3.3. The Sum Rule for Limits.
\begin{equation*}
\lim_{x \to a} [ f(x) + g(x) ] = \lim_{x \to a}f(x) + \lim_{x \to a}g(x).
\end{equation*}
Similarly
Fact 2.3.4. The Difference Rule for Limits.
\begin{equation*}
\lim_{x \to a} [ f(x) - g(x) ] = \lim_{x \to a}f(x) - \lim_{x \to a}g(x).
\end{equation*}
Other basic arithmetic works too: we have
Fact 2.3.5. The Product Rule for Limits.
\begin{equation*}
\lim_{x \to a} [ f(x) \cdot g(x) ] = \lim_{x \to a}f(x) \cdot \lim_{x \to a}g(x)
\end{equation*}
Fact 2.3.6. The Quotient Rule for Limits.
\begin{equation*}
\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\displaystyle\lim_{x \to a}f(x)}{\displaystyle\lim_{x \to a}g(x)},
\quad
\text{so long as } \lim_{x \to a} g(x) \neq 0.
\end{equation*}
Note well: when the right-hand side does not make sense (division by zero), the left-hand side still might! In fact, many of the most important limit calculations are like that.
The Power Rule, and Power Functions.
Using the product rule repeatedly gives
Fact 2.3.7. The Power Rule for Limits.
\begin{equation*}
\lim_{x \to a} [f(x)]^n = [\lim_{x \to a}f(x)]^n \, \text{ for } n \text{ any natural number.}
\end{equation*}
The power rule applied to the simple function \(f(x)=x \) gives the limits of power functions
\begin{equation*}
\lim_{x \to a} x^n = a^n.
\end{equation*}
Limits of Polynomials and Rational Functions.
The rule for constant multiples gives the limit for any monomial \(f(x) = C x^n\text{:}\)
\begin{equation*}
\lim_{x \to a} f(x) = \lim_{x \to a} C \cdot x^n = C \cdot \lim_{x \to a} x^n = C \cdot a^n = f(a).
\end{equation*}
Any polynomial \(p(x)=c_0+c_1 x + c_2 x^2 + \cdots \) is a sum of such monomials, so using the Addition Rule repeatedly gives
\begin{equation*}
\lim_{x \to a} p(x)
= \lim_{x \to a} c_0+c_1 x + c_2 x^2 + \cdots
= \lim_{x \to a} c_0 + \lim_{x \to a}c_1 x + \lim_{x \to a}c_2 x^2 + \cdots
= c_0+c_1 a + c_2 a^2 + \cdots
= p(a).
\end{equation*}
So we have every limit of every polynomial: they are all given simply by evaluating at \(x = a\text{.}\)
Example 2.3.8.
Find a simple strategy for calculating the limit of any rational function, \(\displaystyle\lim_{x \to a} \frac{p(x)}{q(x)} \) where \(p(x) \) and \(q(x) \) are polynomials.
Hint.Use the rules above first for quotients, then for polynomials.
Theorem 2.3.9. The Direct Substitution Property.
For \(f\) a polynomial or rational function and any number \(a \) in its domain, the limit is given by simply substituting \(a \) for \(x \text{:}\)
\begin{equation*}
\lim_{x \to a} f(x) = f(a)
\end{equation*}
We will gradually expand the list of functions with this nice property, called continuity.
Checkpoint 2.3.10.
Evaluate \(\displaystyle\lim_{x \to 2}\frac{x-2}{x^2-4} \text{,}\) using the limit laws and as little algebra as possible.
Theorem 2.3.11. The Root Law for Limits.
\begin{equation*}
\lim_{x \to a} \sqrt[n]{x} = \sqrt[n]{a} \text{ for } n \text{ a positive integer } ( a \gt 0 \text{ is needed if } n \text{ is even).}
\end{equation*}
More generally
\begin{equation*}
\lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\displaystyle\lim_{x \to a}f(x)},
\end{equation*}
this time with \(\displaystyle\lim_{x \to a} f(x) \gt 0 \) needed if \(n \) is even.Proof.
Ignoring the Function Value at \(a\).
Remember that the value of \(f(x) \) for \(x=a \) is irrelevant to the limit as \(x \to a \text{:}\)
If \(f(x) = g(x) \) for \(x \neq a \text{,}\) then \(\displaystyle\lim_{x \to a}f(x) = \lim_{x \to a}g(x) \text{.}\)
This includes the possibility that neither limit exists: both are ``DNE''. Thus only \(x \lt a \) and \(x \lt a \) matter, and it sometimes helps to consider these two cases one at a time.
Using One-sided Limits to Compute Limits.
As seen in the previous section, a limit exists exactly when both one-sided limits exist, and both of then have the same value, in which case the limit has that same value too. Moreover, all the rules seen above are also true for one sided limits.
Checkpoint 2.3.12.
Evaluate \(\displaystyle\lim_{x \to 2} f(x) \) where
\begin{equation*}
f(x) = \left\{ \begin{array}{rl} x^3, & x \lt 2 \\ 3, & x= 2 \\ x^2 + 2x, & x \gt 2 \end{array} \right.
\end{equation*}
Hint: compute each one-side limit. Also, it might helps to sketch the graph first; it often does!
Limits Respect Inequalities.
If \(f(x) \) is no greater than \(g(x)\text{,}\) its limit at any point is not greater either. That is,
\begin{equation*}
\text{If } f(x) \leq g(x), \text{ then } \lim_{x \to a} f(x) \leq \lim_{x \to a} g(x),
\end{equation*}
(so long as both limits exist.)
This inequality idea is particularly useful when a function \(f(x) \) can be squeezed between two simpler functions \(l(x) \) and \(u(x) \) with both of them having the same limit at a point \(a \text{;}\) this situation forces the in-between function \(f(x) \) to have that same limit:
Theorem 2.3.13. The Squeeze Theorem.
If \(l(x) \leq f(x) \leq u(x)\) and \(\displaystyle \lim_{x \to a} l(x) = \lim_{x \to a} u(x) = L \text{,}\)
then \(\displaystyle \lim_{x \to a}f(x) = L\) also.
Checkpoint 2.3.14.
Sketch \(\displaystyle y = f(x) = x^2 \sin \left(\frac{1}{x} \right)\) for \(x \) near 0, and then evaluate \(\displaystyle\lim_{x \to 0}f(x) \text{.}\)
In a while we will use squeezing to show that \(\displaystyle\lim_{x \to 0}\frac{\sin x}{x}=1 \text{,}\) and use that to find the slope at any point on the graph of any trig. function.
Exercises Exercises
Study Calculus Volume 1, Section 2.3 2 , Exercises 83, 85, 89, 91, 93, 97, 107, 111, 119, 121, 127, and 128.
openstax.org/books/calculus-volume-1/pages/2-3-the-limit-lawsopenstax.org/books/calculus-volume-1/pages/2-3-the-limit-laws
