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Section 1.6 Limit laws

Now that we have gained enough intuition about limits to make our first rough defintion, we turn to thinking about the properties that limits have.

We next introduce limit laws. These are not about computing limits directly but instread are tools that we can use to break up a complex limit problem into several (hopefully) simpler problems. This process - breaking hard problems into a sequence of problems we know how to solve and then putting everything together into a single solution - is one of the recurring themes of the course.

We can also think about this in the opposite direction. Over time we will learn how to evaluate limits of more and more functions and types of functions. Limit laws allow us to combine our knowledge of limits of several functions to learn how find the limit of a more complicated function that is built from the ones we know.

To that end we will state two facts: that the limit as \(x\) approaches \(a\) of the function \(f(x)=x\) is equal to \(a\) or,

\begin{equation*} \lim_{x \rightarrow a}x =a, \end{equation*}

and that the limit of a constant function is simply that constant,

\begin{equation*} \lim_{x \rightarrow a}c =c. \end{equation*}

We are going to see how much knowledge of limits we can build with this single known limit and the limit laws.

Let's start by assuming that we know about two limits, that

\begin{equation*} \lim_{x \rightarrow a} f(x)=L \end{equation*}

and

\begin{equation*} \lim_{x \rightarrow a} g(x)=M. \end{equation*}

Generally limit laws tells us what happens when we add, multiply or divide these functions together and take their limits.

Our first law is the Sum law for limits:

\begin{equation*} \lim_{x \rightarrow a} (f(x)+g(x))=\lim_{x \rightarrow a} f(x)+\lim_{x \rightarrow a} g(x) = L+M. \end{equation*}

If we take \(f(x)=g(x)=x\) what does this tell us? Using the law we get

\begin{equation*} \lim_{x \rightarrow a} 2x = \lim_{x \rightarrow a} x+\lim_{x \rightarrow a} x=a+a=2a. \end{equation*}

In fact, if we add \(x\) to itself lots of times we can compute the limit by repeated application of this law:

\begin{equation*} \lim_{x \rightarrow a} kx = \lim_{x \rightarrow a} x+\dots+\lim_{x \rightarrow a} x=a+\dots+a=ka. \end{equation*}

Our second law is the Difference law for limits:

\begin{equation*} \lim_{x \rightarrow a} (f(x)-g(x))=\lim_{x \rightarrow a} f(x)-\lim_{x \rightarrow a} g(x) = L-M, \end{equation*}

and our third law is the Constant multiple law for limits, if \(c\) is a constant, then

\begin{equation*} \lim_{x \rightarrow a} cf(x)=c\lim_{x \rightarrow a} f(x)=cL. \end{equation*}

Some of these laws overlap one another, for example we can put the constant multiple law together with the sum law to get the difference law. Combine the two looks like this:

\begin{equation*} \lim_{x \rightarrow a} (f(x)+cg(x))=\lim_{x \rightarrow a} f(x)+\lim_{x \rightarrow a} cg(x) =\lim_{x \rightarrow a} f(x)+c\lim_{x \rightarrow a} g(x)=L+cM. \end{equation*}

The first equality is true by the sum law while the second is true by the constant multiple law. Letting \(c=-1\) in that sequence of equations gives us the difference law!

Our next law is the Product law for limits:

\begin{equation*} \lim_{x \rightarrow a} (f(x)\cdot(x))=\lim_{x \rightarrow a} f(x) \cdot\lim_{x \rightarrow a} g(x) = LM. \end{equation*}

Now we can build up even more functions from \(f(x)=x\text{.}\) Applying this law once gives:

\begin{equation*} \lim_{x \rightarrow a} x^2=\lim_{x \rightarrow a} x \cdot\lim_{x \rightarrow a} x = a\cdot a=a^2. \end{equation*}

Like we did with the sum rule, we can apply this over and over to get

\begin{equation*} \lim_{x \rightarrow a} x^k=\lim_{x \rightarrow a} x \cdots\lim_{x \rightarrow a} x = a\cdots a=a^k. \end{equation*}

With these tools, we can now make a big jump - we can evaluate the limit of any polynomial! Let's see how using the most generic polynomial:

\begin{equation*} f(x)=a_nx^n+a_{n-1}x^{n-1}+\dots+a_1x_1+a_0. \end{equation*}

If we want to calculate the limit of \(f(x)\) as \(x\) approaches \(a\text{,}\) we first use the sum law to break the larger limit into many limits of the individual terms of the polynomial. Then, using the constant multiple law, we can move the coefficients of the polynomial terms outside the limit. What we are left with is a collection of limits we just figured out how to evaluate - powers of \(x\text{.}\) As the limit of \(x^k\) and \(x\) approaches \(a\) is \(a^k\text{,}\) we can quickly recognize the limit of the polynomial as simply the evaluation of the polynomial at \(x=a\text{.}\)

We've been able to accomplish a lot with just three laws! But we have a few more - the Quotient law for limits

\begin{equation*} \lim_{x \rightarrow a} \frac{f(x)}{g(x)}=\frac{\lim_{x \rightarrow a} f(x)}{\lim_{x \rightarrow a} g(x)} = \frac{L}{M}, \end{equation*}

whenever \(M\neq 0\) and the Power law for limits

\begin{equation*} \lim_{x \rightarrow a} (f(x))^n=(\lim_{x \rightarrow a} f(x))^n=L^n, \end{equation*}

and the Root law for limits

\begin{equation*} \lim_{x \rightarrow a} \sqrt[n]{f(x)}=\sqrt[n]{\lim_{x \rightarrow a} f(x)}=\sqrt[n]{L}. \end{equation*}

The last law is only true for all \(L\) when \(n\) is odd and is true for \(L \ge 0\) when \(n\) is even and \(f(x) \ge 0\text{.}\)

Just as we did before, we can use these laws to help us evaluate more and more complex functions. For example, combining our previous work with the Quotient law for limits we can learn to evaluate the limit of some rational functions. A rational function is just the quotient of two polynomials, which we will call \(p\) and \(q\text{.}\) We know from before that \(\lim_{x \rightarrow a}p(x)=p(a)\) and \(\lim_{x \rightarrow a}q(x)=q(a)\) so if \(q(a)\neq 0\) the Quotient law for limits says

\begin{equation*} \lim_{x \rightarrow a}\frac{p(x)}{q(x)}= \frac{\lim_{x \rightarrow a}p(x)}{\lim_{x \rightarrow a}q(x)}=\frac{p(a)}{q(a)}. \end{equation*}

If we look at our work so far, we might be tempted to think that there's not much to taking limits - just evaluate the function at the point! While there are a large class of functions where this is true, there are many where it is not. As we will see that the solution to our two central questions - the tangent and area problems - rests on being able to evaluate very difficult limits.