B.5 Use Algebra to Find a Limit

As a third and final tactic, let's look at other ways to use algebra to find a limit. These are just "other algebraic moves" — things like expanding a quadratic, or putting terms over a common denominator. Basically: do what you gotta do to keep modifying the expression until you can use Substitution. We have practice problems below for you to use, of course each with a complete solution.

Rather than providing an Example, let's dive in with some Scaffolded Problems. Please give them a try, and use the additional guidance in each Step 2 if you'd like.

Use Algebra to Find a Limit: Scaffolded Exercise #1

Find lim𝑥→0(𝑥−2)2−4𝑥.

Solution.

Step 1. First try substitution:

Show/Hide Step 1 Solution

lim𝑥→0(𝑥−2)2−4𝑥=(2−2)20=00

Because this limit is in the form of 00, it is indeterminate—we don't yet know what it is. We thus have more work to do. So on to Step 2...

Step 2. Use algebra and simplify:

In this case, expand the quadratic and then simplify:

Show/Hide Step 2 Solution

lim𝑥→0(𝑥−2)2−4𝑥=lim𝑥→0(𝑥2−4𝑥+4)−4𝑥=lim𝑥→0𝑥2−4𝑥𝑥=lim𝑥→0𝑥(𝑥−4)𝑥=lim𝑥→0𝑥(𝑥−4)𝑥=lim𝑥→0(𝑥−4)

Step 3. Substitution to finish:

Show/Hide Step 3 Solution

lim𝑥→0(𝑥−2)2−4𝑥=lim𝑥→0(𝑥2−4𝑥+4)−4𝑥=0−4=−4✓

This approach works for essentially the same reason the factoring tactic and the conjugate tactic work:the functions (𝑥−2)2−4𝑥 and 𝑥 −4 are the same, except that the original function is not defined at 𝑥 =0, whereas the rewritten function is.

Use the same types of approaches for the following practice problems.

The Upshot

1. When you try Substitution, if you obtain 00 and have a quadratic (or cubic, or ...) you can expand, or have some fractions you can put over a common denominator, do it. After simplifying, Substitution will probably work.

On the next screen we'll take a quick look at the Squeeze (or "Sandwich") Theorem, which you should at least know about.

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