Question

\(\lim _{\mathrm{x} \rightarrow-8}\left[\left({ }^{3} \sqrt{\mathrm{x}}+2\right) /\\{\sqrt{(1-\mathrm{x})-3\\}]}=?\right.\) (a) \((1 / 8)\) (b) \(-(1 / 2) \quad\) (c) \(+(1 / 4)\) (d) \(-(1 / 8)\)

Short answer

The short answer to the question is: None of the given options are correct. The limit of the given function as x approaches -8 is 0.

Step-by-step solution

Factor out the cube root

First, rewrite the cube root for better visualization: \(\lim _{x \rightarrow-8}\left[\frac{\sqrt[3]{x}+2}{\sqrt{(1-x)-3}}\right]=?\)

Simplify the expression inside the square root

Simplify the expression inside the square root in the denominator: \(\lim _{x \rightarrow-8}\left[\frac{\sqrt[3]{x}+2}{\sqrt{-x-2}}\right]=?\)

Evaluate the limit

Now substitute the limit point, x = -8, into the expression and see if it yields any of the given answer choices: \(\frac{\sqrt[3]{-8}+2}{\sqrt{-(-8)-2}}=\frac{-2+2}{\sqrt{8-2}}=\frac{0}{\sqrt{6}}=0\) The given limit is equal to zero, but none of the answer options match this result. It's possible that there might be some error in the given exercise or answer choices. However, even though our solution does not match any of the provided answer choices, we can confidently state that the limit of the given function as x approaches -8 is 0.

Key concepts

Cubic Root Simplification

Understanding how to simplify cubic roots is a crucial step in evaluating limits of functions that include them. A cubic root, denoted as \(\sqrt[3]{x}\), is the number that, when multiplied by itself three times, gives the number \(x\). For example, the cubic root of -27 is -3 because \( (-3) \cdot (-3) \cdot (-3) = -27 \).

In the context of simplifying expressions, it's important to note that \(\sqrt[3]{x}\) is continuous for all real numbers. This means we can easily substitute values into \(\sqrt[3]{x}\) without worrying about undefined operations, unlike square roots where we cannot take the square root of a negative number without involving complex numbers.

When simplifying cubic roots in an expression, especially when dealing with limits, you should directly substitute the limit value into the cubic root if possible. If the result is an indeterminate form, like \(\frac{0}{0}\) or \(\infty / \infty\), further simplification or the application of L'Hôpital's rule may be necessary.

Limit Evaluation

Limit evaluation is a fundamental concept in calculus that determines the value that a function approaches as the input approaches some value. The notation \(\lim _{x \rightarrow c}f(x)\) represents the limit of the function \(f(x)\) as \(x\) approaches the value \(c\).

When evaluating a limit, you can often substitute the limiting value directly into the expression, provided that it does not result in an indeterminate form. If the direct substitution method does not work, you may need to use other strategies such as factoring, rationalizing, or applying L'Hôpital's rule.

It is also worth noting that limits may exhibit different behaviors. A limit may have a finite value, it might be infinite, or it might not exist at all. Understanding the behavior of the function around the limiting value is critical in correctly evaluating the limit. In the context of cubic roots within limits, direct substitution is typically a strong initial approach since cubic roots are defined for all real numbers.

Expression Simplification

Expression simplification is the process of making a mathematical expression easier to understand and work with. The goal is to transform complicated expressions into simpler or more recognizable forms without changing their value.

This could involve combining like terms, factoring, expanding expressions, rationalizing denominators, or canceling common factors. Simplification not only makes the expressions visually cleaner, but it can also reveal features of the function that are not obvious in its original form, which is especially helpful in limit problems.

When faced with a complex limit problem, the initial simplification of the expression can be the difference between an easily solvable problem and one that seems intractable. By simplifying, we aim to reach a point where direct substitution is possible or an indeterminate form is resolved. For example, to simplify \(\sqrt[3]{x} + 2\), we may look to isolate the cubic root or alter the expression so that we could combine it with other terms to eliminate complex components, like radicals or fractions, in the context of a limiting process.