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Chapter 3: Problem 34

Finding a Limit In Exercises \(19-38,\) find the limit. $$ \lim _{x \rightarrow-\infty} \frac{2 x}{\left(x^{6}-1\right)^{1 / 3}} $$

Short Answer

Expert verified
The limit of the given function as x approaches negative infinity is 0.

Step by step solution

01

Understanding the problem

The given function is given by \(\frac{2x}{{(x^6-1)}^{1/3}}\). We are supposed to find the limit of this function as \(x\) approaches negative infinity.
02

Analyze the behavior at infinity

As x approaches negative infinity, the term \((x^6-1)^{1 / 3}\) also tends to minus infinity. Since the numerator is directly proportional to \(x\), it also tends to minus infinity. So it is an indeterminate form of type \( \frac{-\infty}{-\infty}\). We can further simplify the expression to easier calculate the limit.
03

Simplification

Divide both the numerator and the denominator by \(x^2\). The function becomes \(\frac{2}{{(x^4-\frac{1}{x^2})}^{1/3}}\). Now, as \(x\) tends to negative infinity, the function simplifies to \(\frac{2}{-|x|}\).
04

Final computation

The limit of \(\frac{2}{-|x|}\) as \(x\) approaches negative infinity is \(0\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Indeterminate Forms
When dealing with limits, particularly involving infinity, you might encounter situations that seem undefined or nonsensical at first glance. These are known as indeterminate forms. An example is when both the numerator and the denominator in a fraction approach infinity, like \(\frac{-\infty}{-\infty}\). This indicates that simple substitution won't work, and we need to analyze further to find a clear answer.
  • Indeterminate forms can sometimes be reduced to a simpler form that we can evaluate.
  • This often involves algebraic manipulation, such as factoring or using conjugates.
  • By rewriting the expression, we can uncover the true behavior of the function as it approaches infinity.
Understanding indeterminate forms is critical. It highlights the need to look deeper and not take numerical forms as final results until simplified.
Infinity in Calculus
Infinity represents a concept in calculus, rather than an actual number. When dealing with functions approaching infinity, it's important to note how they behave rather than focusing on specific values. Seeing how a function behaves as it approaches infinity can solve complex limit problems.
  • Consider the direction of approach: Whether a function is approaching positive or negative infinity impacts its behavior.
  • When solving limits, you often assess how the degree of terms in a polynomial relate, which helps determine dominance as \(x\) approaches infinity.
  • Simplifying complex expressions often involves identifying dominant terms that dictate behavior at large values of \(x\).
In the exercise, we observed how both the numerator and denominator approach infinity, initially causing an indeterminate form. Our task was to resolve this to better understand the limit at negative infinity.
Simplification of Expressions
Simplification is essential in dealing with complex mathematical expressions or indeterminate forms. It's the bridge between complicated original forms and understandable results. In the given solution, simplification allowed us to move from the indeterminate form to a limit result of zero. To simplify expressions effectively:
  • Divide both the numerator and denominator by the highest power of \(x\) seen in the denominator.
  • Factor out common terms and reduce fractions where applicable.
  • In cases involving powers or roots, consider the dominant terms as \(x\) approaches infinity, simplifying fractional powers accordingly.
By following these steps, we were able to simplify the function to \(\frac{2}{-|x|}\), clarifying the limit calculation. This process showed how seemingly complex problems could be broken down into manageable steps. Simplification, therefore, not only provides clarity but also guides us towards the solution in an efficient manner.

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Most popular questions from this chapter

True or False? In Exercises \(47-50\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(y=f(x), f\) is increasing and differentiable, and \(\Delta x>0\) , then \(\Delta y \geq d y\)

Verifying a Tangent Line Approximation In Exercises 41 and \(42,\) verify the tangent line approximation of the function at the given point. Then use a graphing utility to graph the unction and its approximation in the same viewing window. $$ \begin{array}{llI}{\text { Function }} & {\text { Approximation }} & {\text { Point }}\\\ {f(x)=\sqrt{x+4}} & {y=2+\frac{x}{4}} & (0,2) \\\\\end{array} $$

Ohm's Law A current of \(I\) amperes passes through a resistor of \(R\) ohms. Ohm's Law states that the voltage \(E\) applied to the resistor is $$E=I R\( \). The voltage is constant. Show that the magnitude of the relative error in \(R\) caused by a change in \(I\) is equal in magnitude to the relative error in \(I .\)

Numerical, Graphical, and Analytic Analysis The cross sections of an irrigation canal are isosceles trapezoids of which three sides are 8 feet long (see figure). Determine the angle of elevation \(\theta\) of the sides such that the area of the cross sections is a maximum by completing the following. (a) Analytically complete six rows of a table such as the one below. (The first two rows are shown.) $$ \begin{array}{|c|c|c|c|c|}\hline \text { Base } 1 & {\text { Base } 2} & {\text { Altitude }} & {\text { Area }} \\ \hline 8 & {8+16 \cos 10^{\circ}} & {8 \sin 10^{\circ}} & {\approx 22.1} \\ \hline 8 & {8+16 \cos 20^{\circ}} & {8 \sin 20^{\circ}} & {\approx 42.5} \\ \hline\end{array} $$ (b) Use a graphing utility to generate additional rows of the table and estimate the maximum cross-sectional area. (Hint: Use the table feature of the graphing utility.) (c) Write the cross-sectional area \(A\) as a function of \(\theta\) . (d) Use calculus to find the critical number of the function in part (c) and find the angle that will yield the maximum cross-sectional area. (e) Use a graphing utility to graph the function in part (c) and verify the maximum cross-sectional area.

Volume and Surface Area The measurement of the edge of a cube is found to be 15 inches, with a possible error of 0.03 inch. (a) Use differentials to approximate the possible propagated (a) Use differentials the volume of the cube. (b) Use differentials to approximate the possible propagated error in computing the surface area of the cube. (c) Approximate the percent errors in parts (a) and (b).
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