Using Change of Variables

Instructor: Catherine Glover

Catherine has a master's degree in Mathematical Biology and teaches math at the college level.

Ever try to find the limit of a function, but get stuck with an undefined solution? Try simplifying the function with a change of variable! Read this lesson to see examples of the change of variables method for finding the limit of a function.

Time to Simplify

If you've ever studied a foreign language, you know that it can seem very complicated at first. Learning vocabulary, grammar, and pronunciation in a totally new language takes a lot of work! But many languages are related, and recognizing the similarities between your native language and the one you're trying to learn can help simplify the process.

We can use a similar process for solving problems in mathematics. Functions can appear very complicated at first, with many different parts, including products, quotients, exponents, and radicals. What's worse, we often need to take the limit of a function where the function itself is undefined, or discontinuous!

Luckily, we can use the change of variables method, in which we define a part of the function as a new variable, to simplify the process of finding the limit of a function. The change of variable method is also often called u-substitution, because the variable we introduce is often called u, and we substitute forms of u in the place of the function's original variable. In this lesson, we'll explore using the change of variables method to find the limits of tricky functions where the function is undefined.

Rational Exponents

Let's try to find the following limit:

Rational function

Looks tricky, right? This is a rational function, so it's undefined when the denominator equals 0, which occurs when x = 27. Because the function is undefined at x = 27, we can't assume that the limit of the function as x approaches 27 is equal to the function at x = 27. Let's try using a change of variable to simplify the function.

Let's introduce the variable u = x1/3. We're changing the variable to u, so we want to replace all instances of x in the function with a form of u. The denominator of the function contains an x. We defined u = x1/3, so if we simply cube both sides, we can figure out that x = u 3.

Since we are changing the variable in the function, we also need to change the variable in the limit. If x is approaching 27, then u = x1/3 is approaching 271/3 = 3.

Now we can substitute our new variable u into the function and limit:

Rational function with u-substitution

Maybe it doesn't seem simpler at first, but if you think back to your algebra course, you should recognize that the denominator is now a difference of cubes, which can easily be factored.

Difference of cubes factored

The numerator and denominator of the rational function share a common factor of u - 3, so that can be canceled out, leaving us with a numerator of 1. Since this new function is defined at u = 3, we know that the limit of the function as u approaches 3 is equal to the value of the function at u = 3. Therefore:

Finding the limit of the simplified rational function

The limit of the original function as x approaches 27 equals the limit of the simplified function as u = x1/3 approaches 3, which is 1/27.

Multiple Radicals

Here's another example of a complicated-looking function that we want to take the limit of:

Rational function with radicals

Again, the function is undefined where we want to take the limit at x = 1. Maybe getting rid of the radicals would help. But if I substitute u = x, I'm still left with the same radicals. So let's try defining a higher power of u that will allow us to get rid of the radicals. If u 2 = x then:

First u-substitution try

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