# Problems solving a limit which results in an indeterminate form

• greg_rack
In summary, the person is having difficulties with solving a limit that ends up in an indeterminate form. They ask for help and provide their attempted procedure. The conversation then leads to using the binomial expansion or a general factorization method to solve the limit. The expert suggests using identities involving differences of square roots or cube roots to solve the problem.
greg_rack
Gold Member
Homework Statement
##\lim_{x \to +\infty}(\sqrt[3]{x^3-4x^2}-x)##
Relevant Equations
none
Hi guys, I am having difficulties in solving this limit.

Below, I'll attach my procedure which ends up in the indeterminate form ##0\cdot \infty##...
How could I solve it?

$$\lim_{x \to +\infty}(\sqrt[3]{x^3-4x^2}-x) \rightarrow \lim_{x \to +\infty}(x\sqrt[3]{1-\frac{4}{x}}-x) \rightarrow \lim_{x \to +\infty}[x(\sqrt[3]{1-\frac{4}{x}}-1)]$$

What is the binomial expansion of ##(1-\frac{4}{x})^{1/3}##?

greg_rack said:
Homework Statement:: ##\lim_{x \to +\infty}(\sqrt[3]{x^3-4x^2}-x)##
Relevant Equations:: none

Hi guys, I am having difficulties in solving this limit.

Below, I'll attach my procedure which ends up in the indeterminate form ##0\cdot \infty##...
How could I solve it?

$$\lim_{x \to +\infty}(\sqrt[3]{x^3-4x^2}-x) \rightarrow \lim_{x \to +\infty}(x\sqrt[3]{1-\frac{4}{x}}-x) \rightarrow \lim_{x \to +\infty}[x(\sqrt[3]{1-\frac{4}{x}}-1)]$$
You can do it using the binomial expansion, as above. Or, you use a general factorisation method using:
$$(f(x)^{1/3} - x)(f(x)^{2/3} + xf(x)^{1/3} + x^2) = f(x) - x^3$$ with ##f(x) = x^3 -4x^2##

greg_rack and Mark44
PeroK said:
You can do it using the binomial expansion, as above. Or, you use a general factorisation method using:
$$(f(x)^{1/3} - x)(f(x)^{2/3} + xf(x)^{1/3} + x^2) = f(x) - x^3$$ with ##f(x) = x^3 -4x^2##
This is the tack I would take. The basic ideas when dealing with a difference of square roots or a difference of cube roots (as in this problem) are these identities:
##(a - b)(a + b) = a^2 - b^2##
##(a - b)(a^2 + ab + b^2) = a^3 - b^3##
##(a +b)(a^2 - ab + b^2) = a^3 + b^3##
In the first identity, a or b stands for the square root of some expression.
In the second and third, a or b stands for the cube root of some expression.

greg_rack and PeroK
Great, thanks!

## 1. What is an indeterminate form in limit problems?

An indeterminate form in limit problems is when the value of the limit cannot be determined just by looking at the expression. It usually occurs when there is a division by zero or when two functions approach different values as x approaches a certain value.

## 2. How do you solve a limit that results in an indeterminate form?

To solve a limit that results in an indeterminate form, you can use various techniques such as factoring, algebraic manipulation, or L'Hopital's rule. These techniques help to simplify the expression and make it easier to evaluate the limit.

## 3. Can all limits with indeterminate forms be solved?

Not all limits with indeterminate forms can be solved. In some cases, the limit may not exist or may require more advanced techniques to solve. It is important to understand the properties and rules of limits to determine if a limit can be solved or not.

## 4. What is L'Hopital's rule and when should it be used?

L'Hopital's rule is a mathematical technique used to solve limits that result in indeterminate forms. It states that if the limit of the ratio of two functions is an indeterminate form, then the limit of the ratio of their derivatives is equal to the original limit. It should be used when other techniques such as factoring or algebraic manipulation are not applicable.

## 5. Are there any common mistakes to avoid when solving limits with indeterminate forms?

Yes, there are a few common mistakes to avoid when solving limits with indeterminate forms. Some of these include forgetting to check for the existence of the limit, applying L'Hopital's rule incorrectly, and not simplifying the expression before evaluating the limit. It is important to double-check your work and be aware of these common mistakes to ensure accurate solutions.

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