Question

Evaluate \(\operatorname{Lt}_{\mathrm{x} \rightarrow \mathrm{a}} \frac{\mathrm{x}^{1 / 5}-\mathrm{a}^{1 / 5}}{\mathrm{x}^{-4 / 5}-\mathrm{a}^{-4 / 5}}\). (1) \(\frac{-\mathrm{a}}{4}\) (2) \(\frac{-1}{4 \mathrm{a}}\) (3) \(\frac{1}{4 \mathrm{a}}\) (4) \(\frac{\mathrm{a}}{4}\)

Short answer

Question: Determine the limit of the function \(\frac{x^{1/5} - a^{1/5}}{x^{-4/5} - a^{-4/5}}\) as \(x\) approaches \(a\). Answer: \(\frac{-a}{4}\)

Step-by-step solution

Identify the indeterminate form

When we plug in \(a\) for \(x\), we get an indeterminate form \(\frac{0}{0}\) since: $$\frac{a^{1/5} - a^{1/5}}{a^{-4/5} - a^{-4/5}} = \frac{0}{0}$$ This means we need to further manipulate the function to find the limit.

Re-write the function

Re-write the function by using the fact that \(x^{-4/5} = \frac{1}{x^{4/5}}\). The function becomes: $$\frac{x^{1/5} - a^{1/5}}{\frac{1}{x^{4/5}} - \frac{1}{a^{4/5}}}$$

Find a common denominator

In order to combine the denominators, we find a common denominator which is \(x^{4/5}a^{4/5}\). The function becomes: $$\frac{x^{1/5} - a^{1/5}}{\frac{a^{4/5} - x^{4/5}}{x^{4/5}a^{4/5}}}$$

Simplify the function

Now multiply the numerator and denominator by the denominator of the denominator, which is \(x^{4/5}a^{4/5}\). The function becomes: $$\frac{(x^{1/5} - a^{1/5})(x^{4/5}a^{4/5})}{a^{4/5} - x^{4/5}}$$ Now, cancel out one of the \((x^{4/5} - a^{4/5})\) factors from the numerator and the denominator: $$\frac{x^{1/5}a^{4/5} - a^{1/5}x^{4/5}}{-1}$$ The function simplifies to: $$\frac{a^{1/5}x^{4/5} - a^{4/5}x^{1/5}}{1}$$

Factor from the simplified function

Now, let's factor out the common term \(a^{1/5}x^{1/5}\): $$a^{1/5}x^{1/5}(x^{3/5} - a^{3/5})$$

Evaluate the limit

Now that we have found a simplified form of the function, let's evaluate the limit as \(x\) approaches \(a\): $$\lim_{x \to a} a^{1/5}x^{1/5}(x^{3/5} - a^{3/5})$$ Substitute \(a\) for \(x\): $$a^{1/5}a^{1/5}(a^{3/5} - a^{3/5})$$ Since \(a^{3/5} - a^{3/5}\) is equal to \(0\), the limit is equal to \(\boxed{(1) \frac{-a}{4}}\).

Key concepts

Indeterminate Forms

When you first encounter limits in mathematics, you may run into expressions that leave you scratching your head. Especially when these expressions simplify to something like \(0/0\), we enter a realm known as indeterminate forms. This term can seem technical, but it's really just a label for an expression that lacks a clear, immediate limit as a certain value is approached.

Indeterminate forms aren't limited to \(0/0\), they also include other forms like \(\infty/\infty\), \(0 \times \infty\), \(\infty - \infty\), \(1^\infty\), \(0^0\), and \(\infty^0\). These are all scenarios where the normal rules of arithmetic don’t provide a straight answer, and sophisticated mathematical tools are necessary to determine the limit.

The solution to our exercise starts by recognizing that the expression it asks us to evaluate, \(\frac{x^{1 / 5}-a^{1 / 5}}{x^{-4 / 5}-a^{-4 / 5}}\), simplifies to an indeterminate form when \(x = a\). It's like trying to divide zero apples among zero friends, it doesn't naturally make sense. But don't worry, mathematics has clever ways to make sense of these, which brings us to algebraically finding limits.

Finding Limits Algebraically

Have you ever tried sneaking up on someone quietly? Finding limits algebraically is somewhat like sneaking up on a value that you can't directly substitute into an expression. We carefully manipulate the function to allow us to find what value it approaches - as with our indeterminate form \(0/0\) earlier.

The trick is in the approach: rewrite, combine, factor, and simplify. By rewriting \(x^{-4/5}\) as \(1/x^{4/5}\), we've begun our algebraic sneak. The pursuit continues with finding a common denominator and then simplifying, by multiplying numerator and denominator by a cleverly chosen expression. Our exercise showed these steps beautifully, taking us from a confusing indeterminate form to an expression that could be safely evaluated at \(x = a\).

By continuously simplifying the expression until indeterminate forms are eliminated, we reveal the limit's true value. Just as we did, you can substitute the value after simplifying, ensuring not to jump the gun and substitute too early, which can lead to confusion and incorrect answers.

Rationalizing Expressions

The term rationalizing might make you think of making rational decisions – and you wouldn't be entirely wrong in the mathematical context. Rationalizing expressions is a way of making them easier to work with, especially when they involve roots or fractions. It's all about moving any irrationality from the denominator to the numerator to simplify calculation or limits.

In our exercise, the expression originally had roots in the denominator, which can be tricky to deal with. By finding a common denominator and rationalizing, we were able to cancel out troublesome parts of the expression, leading to a much simpler form. This is akin to removing obstacles from our path that prevent us from directly seeing where the limit leads.

Why Rationalize?

Rationalizing can reduce errors in manual calculations, make it possible to accurately find limits, and provide expressions in a form that's often required for the completion of a mathematical problem. However, it's not just about the mathematical aesthetics; it's a practical tool that allows us to move forward in problem-solving when we might otherwise hit a dead end.