We want to evaluate limits for which the Limit Laws do not apply.
In the last section we computed limits using continuity and the limit laws. What about limits that cannot be directly computed using these methods? Consider the following limit, Here in light of this, you may think that the limit is one or zero. Not so fast. This limit is of an indeterminate form. What does this mean? Read on, young mathematician.
Let’s finish the example with the function above.
Let’s consider a few more examples of the form .
What can be done? We can hope to be able to cancel a factor going to out of the numerator and denominator. Since is a factor going to in the numerator, let’s see if we can factor a out of the denominator as well.
The limit of the denominator is: Our limit is therefore of the form and we can probably factor a term going to out of both the numerator and denominator. When looking at the denominator, we hope that this term is . Unfortunately, it is not immediately obvious how to factor an out of the numerator. So, we should first simplify the complex fraction by multiplying it by this will allow us to cancel immediately
Now we will multiply out the numerator. Note that we do not want to multiply out the denominator because we already have an factored out of the denominator and that was the goal.
Now, we can see that the limit of “our function” is equal to the limit of a rational function . This rational function is continuous on its domain, and, therefore, at . Hence
We will look at one more example.
We will use an algebraic technique called multiplying by the conjugate. This technique is useful when you are trying to simplify an expression that looks like It takes advantage of the difference of squares rule In our case, we will use and . Write
All of the examples in this section are limits of the form . When you come across a limit of the form , you should try to use algebraic techniques to come up with a continuous function whose limit you can evaluate.
Notice that we solved multiple examples of limits of the form and we got different answers each time. This tells us that just knowing that the form of the limit is is not enough to compute the limit. The moral of the story is
Limits of the form can take any value.
A form that gives information about whether the limit exists or not, and if it exists gives information about the value of the limit, is called a determinate form.
Finally, you may find it distressing that we introduced a form, namely , only to end up saying they give no information on the value of the limit. But this is precisely what makes indeterminate forms interesting… they’re a mystery!