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How to Solve Limits with Basic Algebra

When substitution doesn’t work in the original limit function — usually because of a hole in the function — you can often use some algebra to manipulate the function until substitution does work (it works because your manipulation plugs up the hole).

You can try various things from Algebra I, like adding or subtracting fractions, multiplying or dividing fractions, canceling, or some other form of simplification. Here’s an example:

  1. Try substitution (always try substitution as your first step).


    On to plan B.

  2. Simplify the complex fraction (that’s a big fraction that contains little fractions) by multiplying the numerator and denominator by the least common denominator of the little fractions, namely 4(x + 4).

  3. Now substitution works.


That’s the limit.

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