2.5: Limits Involving Radical Functions

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There are many problems that will involve taking the nth root of a variable expression, so it is natural that there may sometimes be a need to find the limit of a function involving radical expressions, using square or cube roots, or other roots. Do you think that finding the limit of a function involving radicals would be any different than finding the limit of polynomial or rational functions? Can you think of any ways that radicals might present different problems than polynomials?

Limits with Radical Functions

When evaluating a limit involving a radical function, use direct substitution to see if a limit can be evaluated whenever possible. If not, other methods to evaluate the limit need to be explored.

Take the following function

Screen Shot 2020-09-17 at 4.28.15 PM.png

Screen Shot 2020-09-17 at 4.28.40 PM.png

Therefore,

Screen Shot 2020-09-17 at 4.29.21 PM.png

which could have been determined by directly evaluating f(x) at x=9, i.e., by using direct substitution.

Now, find

Screen Shot 2020-09-17 at 4.30.06 PM.png

In both of the above cases, direct substitution could be used to evaluate the limits and there is no need for alternative methods.

Take a look at the function

Screen Shot 2020-09-17 at 4.32.48 PM.png

First we notice that we should exclude x=−⁵/₇ in any evaluation. Using direct substitution to find the limit results in the indeterminate form ^∞/_∞. To transform the radical expression to a better form, use the fact that the value of x is going to larger and larger positive values. This allows the following:

Screen Shot 2020-09-17 at 4.37.55 PM.png

Therefore,

Screen Shot 2020-09-17 at 4.38.40 PM.png

Now, find

Screen Shot 2020-09-17 at 4.39.06 PM.png

The solution to evaluating the limit at negative infinity is similar to the above approach except that x is always negative.

Screen Shot 2020-09-17 at 4.40.03 PM.png

Therefore.

Screen Shot 2020-09-17 at 4.45.28 PM.png

So far, you have been able to find the limit of rational functions using methods shown earlier. However, there are times when this is not possible. Take the function

Screen Shot 2020-09-17 at 4.47.58 PM.png

Find

Screen Shot 2020-09-17 at 4.48.17 PM.png

Using direct substitution to find the limit results in the indeterminate form 00. In order to evaluate the limit, we need to transform the expression to remove the indeterminate form. This is accomplished by using the relationship for the difference of squares of real numbers: x²−y²=(x+y)(x−y).

We then rewrite and simplify the original function as follows:

Screen Shot 2020-09-17 at 4.59.44 PM.png

Use the difference of squares factoring to remove the 0 in the denominator.

Hence

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Now, find the end behavior of the same function, i.e. find

Screen Shot 2020-09-17 at 5.01.34 PM.png

As x increases to large positive values, the function takes on the indeterminate form ^∞/_∞. The transformation above can also be used to evaluate the limit (Approach 1), as well as the technique used in evaluating rational functions (Approach 2).

Screen Shot 2020-09-17 at 5.06.29 PM.png

Hence

Screen Shot 2020-09-17 at 5.49.56 PM.png

Finally, find

Screen Shot 2020-09-17 at 6.04.41 PM.png

The solution to this problem is that the limit does not exist because the domain of h(x) does not include x<0.

Examples

Example 1

Earlier, you were asked if the methods for evaluating limits involving polynomials and rational functions can be used to find the limits of radical functions. Some of the methods do work for radical functions. The use of direct substitution is a common method. Transforming indeterminate or undefined forms by finding and canceling common factors in the numerator and denominator, or factoring and simplifying the highest degree powers of variables represent common approaches.

One of the noteworthy differences between polynomial and radical functions is that the domain of polynomials can include all real values of the independent variable, but the domain of radical functions, e.g., x√, is restricted.

Example 2

Find

Screen Shot 2020-09-22 at 5.52.44 PM.png

Using direct substitution to find the limit of the function results in the indeterminate form ⁰/₀. To transform the radical expression to a better form, do the following:

Screen Shot 2020-09-22 at 6.10.50 PM.png

…Rationalize the numerator: multiply by the conjugate of the numerator

Therefore,

Screen Shot 2020-09-22 at 6.22.13 PM.png

Review

Find each of the following limits if they exist.

Screen Shot 2020-09-22 at 6.22.42 PM.png

Review (Answers)

To see the Review answers, open this PDF file and look for section 2.6.

Vocabulary

Term	Definition
indeterminate	In mathematics, an expression is indeterminate if it is not precisely defined. There are seven indeterminate forms: ⁰/₀,0⋅∞,^∞/_∞,∞−∞,0⁰,∞⁰, and 1^\infty.
limit	A limit is the value that the output of a function approaches as the input of the function approaches a given value.
radical function	Radical functions are functions which contain nth roots of variable expressions.

Additional Resources

Video: Limits at Infinity

Practice: Limits Involving Radical Functions

Real World: Maverick Surfers