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.

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 Problem #1

\[ \lim_{x \to 0} \,\dfrac{(x-2)^2 – 4}{x} = \dfrac{(2-2)^2}{0}=\dfrac{0}{0} \]
Because this limit is in the form of $\dfrac{0}{0}$, it is indeterminate—we don’t yet know what it is. We thus have more work to do. So on to Step 2…

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Step 2. Use algebra and simplify. In this case, expand the quadratic and then simplify:

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

Use algebra to find a limit: Scaffolded Problem #2

\[ \lim_{x \to 4} \, \frac{x^{-1} – 4^{-1}}{x-4} =\dfrac{4^{-1} – 4^{-1}}{0} =\dfrac{0}{0} \]
Because this limit is in the form of $\dfrac{0}{0}$, it is indeterminate—we don’t yet know what it is.

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Step 2. Use algebra and simplify. It’s hard to see what’s going on with the negative exponents, so first write the numerator-terms as fractions. Then put them over a common denominator. Finally, as usual, simplify.

Now use the same types of approaches for the following practice problems.

Practice Problems: Use Algebra to Find a Limit

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When you try Substitution, if you obtain $\dfrac{0}{0}$ 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. Questions? Comments? Join the discussion over on the Forum!