Box "substitution"

Substitution rarely solves problems by itself, but it is used to simplify them and to move troubles from one place to another. The basic idea is as follows: We have an expression that is more complicated than we would like and we want to make it look simpler by taking a part of it and denoting it using some letter - a new variable.

We start with a very simple example: Find

Since the expression 3x² appears every time we have x, we can give this expression a new name, say y. We get a new limit:

Although it did not help substantially since the problem is essentially the same (so the methods to solve it would also be essentially the same), this new limit looks much easier and calculations will be most likely simpler.

Note that x disappeared from the problem entirely. That is one of the basic rules of substitution. This includes the place under the "lim" symbol, where we changed also the limit point. The new limit point was obtained from the old one a = 2 using the formula y = 3x². This is the second basic rule: Once we decide on the formula for replacement, all other changes we do can be only done using this formula or other formulas derived from it.

Now we summarize how we do substitution.

Substitution:
Step 1. Decide which expression g(x) should be substituted for, set up the basic substitution equality y = g(x).
Step 2. Use the basic substitution equality to "change the variable", that is, replace all appearances of x in the limit by appropriate formulas with y. In the expression in the limit we use the basic substitution equality and equalities that can be derived from it. Under the symbol "lim" we replace x with y and then change the limit point, again using the basic substitution equality.

Notation:

Usually the new limit is of the same type as the original one, so substitution rarely gives an answer, but it can help in obtaining it. In particular, l'Hospital's rule requires differentiation and derivatives of simpler expressions are usually nicer. A nice example is this problem in Solved Problems. Sometimes this simplification can be so crucial that it makes a difference between being and not being able to solve the problem, see this problem in Solved Problems.

Sometimes substitution is not crucial but simply convenient. A trick I like is this. When evaluating expressions with powers and similar terms at infinity, we can use intuitive evaluation. We can also use it at negative infinity, but there it can be tricky.

Example:

It is easy to forget that the square root of x² is |x|; if you put x instead, you get the wrong answer 1. When I am not sure concerning negative infinity, I use substitution to change the limit point into plus infinity.

For more insight, see the general note on substitution.

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