# Evaluating Limit with Radical: Can You Help?

• Precursor
In summary, for the given limit, you can factor the numerator as a difference of cubes and then use the conjugate to solve for the limit. For the second question, you can use the idea of multiplying by 1 and factoring to turn the numerator into a difference of fourth roots, which may help in solving for the limit.
Precursor
Evaluate if the following limit exists:

$$lim_{x\rightarrow1}\frac{\sqrt[3]{x}-1}{x-1}$$

My work:

$$lim_{x\rightarrow1}\frac{\sqrt[3]{x^{2}}-1}{(x-1)(\sqrt[3]{x}+1)}}$$

I did the conjugate, but I'm still left with a radical in the numerator, and I can't seem to factor any further. Can someone help me out?

You have the right idea, but haven't quite carried it off. The radical term in your numerator is x^(2/3) != x^(1/3) in the original limit expression.

x - 1 can be thought of as the difference of cubes, as $(\sqrt[3]{x})^3 - 1^3$.

A difference of cubes can be factored like so:
$$(a^3 - b^3) = (a - b)(a^2 + ab + b^2)$$.

Thanks, that helped for that question. But how about a question like the following(we have not yet been taught the difference of fourth root):

$$lim_{x\rightarrow0}\frac{\sqrt[4]{1+x}-1}{x}$$

Could I do a difference of squares and then the conjugate?

How about if you multiplied by 1 (which is always legal)?

The 1 I am thinking of looks like
$$\frac{(u + 1)(u^2 + 1)}{(u + 1)(u^2 + 1)}$$

where $$u = \sqrt[4]{1 + x}$$.

The idea is that you have something that looks like u - 1 in the numerator, and I want to turn it into u^4 - 1. The other factors to make this happen are (u + 1) and (u^2 + 1).

No guarantees that this will work, but it might make it so that some things cancel so that you don't get zero in both the numerator and denominator.

## 1. What is a limit with a radical?

A limit with a radical is a mathematical concept where we are trying to find the value of a function as it approaches a certain point, or "limit," while also containing a square root or other radical expressions.

## 2. How do I evaluate a limit with a radical?

To evaluate a limit with a radical, you can use algebraic techniques such as factoring, simplifying, and rationalizing the expression. You can also use graphing or numerical methods to estimate the limit.

## 3. What is the importance of evaluating limits with radicals?

Evaluating limits with radicals is important in many areas of math and science, such as calculus, physics, and engineering. It allows us to understand the behavior of functions and make predictions about their values at certain points.

## 4. Can you provide an example of evaluating a limit with a radical?

Sure, let's say we have the function f(x) = √(x+1) and we want to find the limit as x approaches 3. We can simplify the expression to f(x) = √(x+1) = (x+1)^(1/2). Then, we can plug in x=3 to get √(3+1) = √4 = 2. Therefore, the limit is equal to 2.

## 5. Are there any special cases when evaluating a limit with a radical?

Yes, there are a few special cases to consider when evaluating limits with radicals. For example, if the limit involves a radical with an even index, the function must be defined at the limit point and the limit must exist. Another special case is when the limit involves a radical with an odd index, in which case the limit exists even if the function is not defined at the limit point.

• Precalculus Mathematics Homework Help
Replies
8
Views
922
• Precalculus Mathematics Homework Help
Replies
17
Views
919
• Precalculus Mathematics Homework Help
Replies
12
Views
933
• Precalculus Mathematics Homework Help
Replies
10
Views
878
• Precalculus Mathematics Homework Help
Replies
25
Views
890
• Precalculus Mathematics Homework Help
Replies
11
Views
1K
• Precalculus Mathematics Homework Help
Replies
4
Views
1K
• Precalculus Mathematics Homework Help
Replies
4
Views
550
• Precalculus Mathematics Homework Help
Replies
6
Views
1K
• Precalculus Mathematics Homework Help
Replies
6
Views
1K