In this explainer, we will learn how to use the direct substitution method to evaluate limits.
Recall that we are given a function defined near , and seek a number so that the closer gets to , the closer the values get to . Such a number need not even exist. When it does, we write
The substitution method says that which obviously requires that the function be defined at . So we cannot use this to find , just because the function is not defined at .
Even if makes sense, substitution can still fail, depending on what the function is. Consider
Now , because satisfies the first condition: . In spite of this, we cannot say that because of the piecewise definition and the fact that this definition gives different one-sided limits to the left and right of the point , as clear from the graph.
We see that while , meaning that the limit does not exist.
On the positive side, we list when we can apply substitution.
Conditions When Substitution Works
Given a function and a point on the number line, where is defined, we can set the limit as approaches to be just under any of the following conditions:
- is a polynomial, which includes any constant function.
- is rational function which is defined at . In other words, , with polynomials, and .
- belongs to one of the “standard” classes of functions: trigonometric, exponential, and logarithmic.
- is a power function, so with any real number. This includes functions like since this is just .
- is a constant multiple, sum, difference, product, or quotient of functions for which substitution works.
- is composed of functions for which substitution works, so , where permits substitution at and allows substitution at .
Although this is quite a large number, we can use these results even when the function only has a piecewise definition, with the proper choice of the point .
The figure shows the same function above and . Since , we have and, in this case, it is still the case that because the function agrees with another function near , where belongs to the list above.
- Which function ? The polynomial function .
- What does “near” mean? This can be interpreted to mean on the interval which contains our point . On this interval, .
- How does differ from ? The difference is that no matter what interval is chosen around 0, there is no single polynomial that defines on that interval.
Example 1: Finding the Limit of a Polynomial by Substitution
The limit is the value of at because this is a polynomial function:
Example 2: Finding the Limit of Root Functions at a Point by Direct Substitution
The function is a composition of functions where
Since , is defined and equals . As a composition of functions that permit substitution, it follows that
A more subtle example is the following.
Example 3: Finding the Limit of a Function Involving Absolute Values
Given , find .
One way to think of the absolute value function is meaning that it is a composition of functions suitable for using substitution to find limits.
The function is defined for any real number, including , and is a difference of and , both of which are compositions of the absolute value function and polynomial functions.
So we can evaluate the limit using substitution: