B.1 Limit Laws

Calculus Home Limits & Continuity B. Calculating Limits B.1 Limit Laws

On this screen we’re going to start to develop the tools you need to find a limit quickly, staring with “limit laws.” We’ll of course provide practice problems, with complete solutions, for you to use.

In the preceding Section you built a strong understanding of the concept of “limit,” mostly by using interactive Desmos graphs as a tool. You of course won’t always have such a tool available, nor will you want to stop to graph each function you encounter to be able to find a limit at a particular point. So you need a new set of skills to find, quickly, the limit for a given function at a specified point.

We begin here by establishing some basics by looking at “Limit Laws.” These are the official “laws” or “properties” that let us do things like find the limit of two functions added together, or that are multiplied together, and so forth. To state the upshot bluntly: this all works exactly the way you think it should, as you’ll see. While we need to formalize the rules you’ll use frequently, nothing here is likely to surprise you.

The limit of the sum of two functions
As a first example, let’s consider two particular functions f and g that have the limits as $x \to 3$
\[\lim_{x \to 3}f(x) = 2 \quad \text{and}\quad \lim_{x \to 3}g(x) = 6\] That is, for f, you can get as close as you’d like to the output-value of 2 by being sufficiently close to $x=3.$ And for g, you can be as close as you’d like to be the output-value of 6 by being sufficiently close to $x=3.$

Now we ask you to think about the sum of those two functions, $f(x) + g(x).$ Specifically, we ask you to consider the limit as $x \to 3$ of those two functions added together. You would say $\displaystyle{\lim_{x \to 3}[f(x) + g(x)] =\; … ?}$

First illustrating the limit law for the sum of two functions: Left figure shows two curves. The first, y=f(x), has the point on it (3,2) highlighted. The second, y=g(x), has the point (3, 6) highlighted. The right figure shows the sum of the two functions, y = f(x) + g(x), and has the point (3,8) highlighted.

You probably thought $\displaystyle{\lim_{x \to 3}[f(x) + g(x)] = 8,}$ and if so you’re exactly right: you can get as close to the output value of 8 for $f + g$ as you’d like by being sufficiently close to $x=3.$ The left-hand figure above shows the two separate functions, f and g, and their respective limits as $x \to 3.$ The right-hand figure shows the sum of the functions, $y = f(x) + g(x)$: adding the curve and the line in the left figure together produces the curve on the right. And as you can see,

\begin{align*}
\lim_{x \to 3}[f(x) + g(x)] &= \lim_{x \to 3}f(x) + \lim_{x \to 3}g(x) \\[8px] &= 2 + 6 = 8
\end{align*}
In words, we can interpret the left-hand side of the preceding equation as the “limit of the sum-of-two-functions,” which then equals the “sum-of-the-limits of the two functions.”
Easy, right?

More generally, if each of the functions f and g has a limit at $x=a,$ then
\[\lim_{x \to a}[f(x) + g(x)] = \lim_{x \to a}f(x) + \lim_{x \to a}g(x) \]

All of the Limit Laws (Properties, Rules)

Let’s name the preceding expression the “Limits Sum Rule,” and then list it with the other similar rules:

Limit Laws (Properties, Rules)
\begin{align*}
&\;1. \;\lim_{x \to a}[f(x) + g(x)] = \lim_{x \to a}f(x) + \lim_{x \to a}g(x) && \text{[Limits Sum Rule]}\\[8px] &\;2. \;\lim_{x \to a}[f(x) – g(x)] = \lim_{x \to a}f(x) – \lim_{x \to a}g(x) && \text{[Limits Difference Rule]}\\[8px] &\;3. \;\lim_{x \to a}cf(x) = c \lim_{x \to a}f(x) && \text{[Limits Constant Multiple Rule]}\\
&\quad\text{ where $c$ is a constant}\\[8px] &\;4.\; \lim_{x \to a}[f(x)g(x)] = \lim_{x \to a}f(x) \cdot\lim_{x \to a}g(x) && \text{[Limits Product Rule]} \\[8px] &\;5.\; \lim_{x \to a}\dfrac{f(x)}{g(x)} = \dfrac{\displaystyle{\lim_{x \to a}f(x)}}{\displaystyle{\lim_{x \to a}g(x)}}, \quad (\text{for }\displaystyle{\lim_{x \to a}g(x) \ne 0}) && \text{[Limits Quotient Rule]}\\[8px] &\;6. \; \lim_{x \to a}[f(x)]^n = \left[\lim_{x \to a}f(x) \right]^n && \text{[Limits Power Rule]} \\[8px] &\;7. \; \lim_{x \to a}\sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a}f(x)} && \text{[Limits Root Rule]}
\end{align*}

Those rules may seem abstract, so we’ll say again: you can safely add, multiply, divide, …, values just as you automatically would.

The following questions illustrate.

Limits Laws Question 1

Use the graphs of the functions f and g to find the requested limits.

Limits Laws Question 2

Find the following values given
\[\lim_{x \to 0}h(x) = -1 \quad \text{and} \quad \lim_{x \to 0}k(x) =27\]

You may have homework or an exam questions that require you to write out step-by-step which limit property you’re using as you go from one line to the next. If so, we strongly suggest that you practice so you’re used to the formalism before you have to do it on an exam, since you’ll lose points if you don’t get everything just-so. Your instructor will show you what they expect if this is the case.

For us, we’re going to simply make use of the limit laws (without calling any extra attention to the fact that we’re doing so) and move on to the various techniques you’ll come to recognize you need in order to find a function’s limit. We start on the next screen with the most straightforward of these: using “substitution” to find a limit.

The Upshot

When finding limits, the limit of a sum-of-functions equals the sum of the limits-of-the-functions. Similar rules hold for the difference, product, quotient of functions, and so forth.

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