What are the limit rules?
Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher)
That’s exactly what you’re going to learn in today’s calculus lesson.
Let’s jump right to it!
Background
So did you know that evaluating a limit algebraically simply means to “plug in” a value and simplify?
Previously, we utilized graphs to solve limits.
More specifically, we could discern what the y-value was approaching when we approach a specific value along the x-axis from both the left and the right.
But graphs are not the only ways to calculate limits.
In fact, as most textbooks will tell you, we can evaluate limits via 4 different methods:
- Graphically
- Numerically
- Analytically
- Algebraically
But let me tell you a little secret — there are actually only 3 different methods, as analytically and algebraically are synonyms.
So since textbooks can be tricky at times, what are we left with?
- Graphically – Evaluate using a graph.
- Numerically – Either use a graph or create a table of values.
- Analytically – Use algebra.
Alright! That’s great!
And to make things even better, as previously stated, all we have to do to evaluate a limit algebraically is substitute in a value and simplify.
Not so bad eh?
Limits Rules For Calculus
But there are some important techniques for calculating limits that we want to explore, as they are fundamental to your success.
Limit Laws — Calculus
Now, these limit laws may seem intimidating at first, but they’re quite helpful and straightforward to use.
Example – Using The Rules
For instance, suppose we are given the following graph of functions f and g, and we are asked to find the following limit:
\begin{equation}
\lim _{x \rightarrow-2}\left[f(x)^{3}+5 g(x)\right]
\end{equation}
Evaluate The Limit Using Limit Rules
See, that’s all there is to it!
Direct Substitution Rule
Or as I like to call it — the “Plug It In” law!
Direct Substitution — Limits
And as Math is Fun nicely states, “evaluating” just means to find the value of something. And what’s the easiest way for us to find the value of a function at the desired number?
Substitute that number into the function and simplify. We’ve been doing it for years, and that’s all there is to it!
A Few More Examples
Let’s look at two more problems so you can see how painless evaluating limits can be.
Problem #1
\begin{equation}
\lim _{x \rightarrow4}\left[x^{2}+7 x-9\right]=(4)^{2}+7(4)-9=35
\end{equation}
Problem #2
\begin{equation}
\lim _{x \rightarrow-3}\left[\frac{x^{2}-6 x+1}{8+5 x}\right]=\frac{(-3)^{2}-6(-3)+1}{8+5(-3)}=\frac{28}{-7}=-4
\end{equation}
Yep, it’s that easy!
Summary
To calculate a limit, all you have to do is plug in the number and simplify.
Now, I would be remiss in not giving you a healthy warning — there may be times when we don’t receive the desired outcome after plugging in a number. This is called an Indeterminate Form, and we’ll discuss how to tackle these types of questions in our next lesson.
But for now, let’s jump on in and evaluate limits algebraically!
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