More Limits by Factorization: An Introduction to Calculus
In our previous article, we introduced the concept of limits by factorization and provided an intuitive yet rigorous explanation for the method.
You’ve already seen an example in our last article. Let’s continue discussing limits by factorization and rationalization:
Many limits cannot be found via direct substitution because they lead to the indeterminate form, so when finding limits, we revert back to an algebraic way of problem solving seen in Algebra.
Let’s try a similar example. Take a look at the expression:
If I asked you to factor, it would be easy. (x²+x-6) factors to (x+3)(x-2), and the (x-2) can then be eliminated. But what if I asked you to find the limit as x→2?
For reference, here is the graph of the function. Take a moment and see if you can determine the limit graphically:
Algebraically, we can employ the same method as in our previous article:
From our previous article, we proved that we can eliminate terms like (x-2) when finding limits, just like in a regular function, even though the function technically changes when we do that.
Once we eliminate (x-2), we are left with:
Simply plug in 2 where we see x (remember that we are trying to find the value that x “approaches”), and that will be our answer.
Limits have some special properties we will discuss in a later article, but I think limits by factoring and rationalization aren’t hard for the reason that they behave so much like a regular function in these scenarios.
Of course, difficulty is in the eye of the beholder, and while the math itself isn’t particularly hard, the intuition and understanding that comes with the topic is.
Treat the limit as a regular algebraic operator, instead of as some mystical symbol you have to work around.
As long as you keep in mind the difference between limits and continuity as discussed earlier, you should have no problems treating them this way for now, and solving them easily using the various tools of algebra.
Let’s try another one, using the same process:
- Factor out like terms
2. Eliminate x²
3. Expand via difference of squares
4. Eliminate (x+1)
5. Plug in (-1)
6. Simplify. -6 is the final answer.
Hopefully that wasn’t too bad. The entire time, we treated the limit like a regular function, and plugged in x at the end to get to our answer. Note that direct substitution would not work because plugging in (-1) would lead to the indeterminate form.
Hopefully solving this question step-by-step makes the slightly awkward concept of limits by factorization more easily accessible.
They are our first tool in the great workshop of limit problem-solving, and we will learn of others in articles yet to come.
This is the sixth installation in a series of articles attempting to explain the entirety of Calculus AB to a mainstream audience. It is designed to be easily understood, without resorting to dumbing down complex and nuanced topics. A full article list can be found on my profile.
