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Chapter 4: Problem 19

\(5-64\) Find the limit. Use I'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{x \rightarrow \infty} \frac{e^{x}}{x^{3}}$$

Short Answer

Expert verified
The limit is \( \infty \).

Step by step solution

01

Determine if the form is indeterminate

Evaluate the expression \( \frac{e^x}{x^3} \) as \( x \to \infty \). Exponential \( e^x \) grows much faster than polynomial \( x^3 \), but let's first confirm if it is in an indeterminate form such as \( \frac{\infty}{\infty} \). As \( x \to \infty \), both \( e^x \to \infty \) and \( x^3 \to \infty \). Thus, this is an \( \frac{\infty}{\infty} \) form.
02

Apply L'Hospital's Rule

Since the limit is of the form \( \frac{\infty}{\infty} \), we can use L'Hospital's Rule. Differentiate the numerator and denominator separately. The derivative of \( e^x \) is \( e^x \), and the derivative of \( x^3 \) is \( 3x^2 \). Therefore, the limit becomes: \[ \lim_{{x \to \infty}} \frac{e^x}{x^3} = \lim_{{x \to \infty}} \frac{e^x}{3x^2}. \] Again, this is \( \frac{\infty}{\infty} \), so apply L'Hospital's Rule again.
03

Apply L'Hospital's Rule Again

Differentiate the numerator and denominator again. The derivative of \( e^x \) remains \( e^x \), and the derivative of \( 3x^2 \) is \( 6x \). Therefore, the limit becomes: \[ \lim_{{x \to \infty}} \frac{e^x}{3x^2} = \lim_{{x \to \infty}} \frac{e^x}{6x}. \] Still indeterminate \( \frac{\infty}{\infty} \), so apply L'Hospital's Rule once more.
04

Final Application of L'Hospital's Rule

Differentiating again, the derivative of \( e^x \) is \( e^x \), and the derivative of \( 6x \) is \( 6 \). Thus, the limit becomes: \[ \lim_{{x \to \infty}} \frac{e^x}{6x} = \lim_{{x \to \infty}} \frac{e^x}{6}. \] Since the numerator \( e^x \to \infty \) while the denominator \( 6 \) is constant, the limit diverges to \( \infty \).
05

Conclusion

The limit \( \lim_{{x \to \infty}} \frac{e^x}{x^3} \) evaluates to \( \infty \) since exponential growth outpaces polynomial growth.

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Key Concepts

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

Limits in Calculus
In calculus, limits help us understand the behavior of functions as they approach specific points or infinity. Limits are foundational in calculus, especially when evaluating how functions behave when approaching a boundary. In the problem, we are analyzing the limit as \( x \to \infty \) for the function \( \frac{e^x}{x^3} \). This expression shows behavior at infinity and allows us to apply L'Hospital's Rule, a powerful tool in calculus.

**When to Use L'Hospital's Rule:**
  • L'Hospital's Rule applies when the limit is of an indeterminate form such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).
  • It involves differentiating the numerator and denominator until the indeterminate form is resolved.
Applying limits and L'Hospital's Rule can simplify complex expressions, making it easier to understand how functions behave under extreme conditions such as at infinity.
Indeterminate Forms
Indeterminate forms occur in calculus when an expression does not have a clear arithmetic result. Common forms are \( \frac{\infty}{\infty}, \frac{0}{0}, \infty - \infty, 0 \times \infty, \text{and} \infty^0 \). Each represents a scenario where algebraic manipulation or calculus tools, like L'Hospital's Rule, are necessary.

**Characteristics of Indeterminate Forms:**
  • The two most common are \( \frac{0}{0} \) and \( \frac{\infty}{\infty} \).
  • These forms signal that direct substitution or evaluation isn't possible.
In our exercise, the form \( \frac{\infty}{\infty} \) arises when both \( e^x \) and \( x^3 \) grow towards infinity as \( x \to \infty \). L'Hospital's Rule can be repeatedly applied until an indeterminate form is resolved, allowing us to find the limit or conclude divergence.
Exponential Growth vs Polynomial Growth
Exponential and polynomial growth differ significantly, especially as \( x \to \infty \). Exponential growth functions like \( e^x \) increase far more rapidly than polynomial functions like \( x^3 \). Here's why:

**Comparison:**
  • Exponential functions have the form \( e^x \), where 'e' is Euler's number, and their growth is proportional to their size, leading to rapid increases.
  • Polynomial functions such as \( x^3 \) grow at a rate determined by their highest power.
  • At larger values of \( x \), exponentials will always eventually exceed polynomials in growth.
In the given exercise, \( e^x \) clearly outpaces \( x^3 \) as \( x \to \infty \). This understanding allows us to predict that the limit \( \lim_{x \to \infty} \frac{e^x}{x^3} \) diverges to infinity because the exponential growth dominates.

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Most popular questions from this chapter

A rain gutter is to be constructed from a metal sheet of width 30\(\mathrm { cm }\) by bending up one-third of the sheet on each side through an angle \(\theta\) . How should \(\theta\) be chosen so that the gutter will carry the maximum amount of water?

Describe how the graph of \(f\) varies as \(c\) varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when c changes. You should also identify any transitional values of \(c\) at which the basic shape of the curve changes. \(f(x)=x^{4}+c x^{2}\)

A manufacturer has been selling 1000 flat-screen TVs a week at \(\$ 450\) each. A market survey indicates that for each \(\$ 10\) rebate offered to the buyer, the number of TVs sold will increase by 100 per week. (a) Find the demand function. (b) How large a rebate should the company offer the buyer in order to maximize its revenue? (c) If its weekly cost function is \(C ( x ) = 68,000 + 150 x\) , how should the manufacturer set the size of the rebate in order to maximize its profit?

If an initial amount \(A_{0}\) of money is invested at an interest rate \(r\) compounded n times a year, the value of the investment after t years is $$A=A_{0}\left(1+\frac{r}{n}\right)^{n t}$$ If we let \(n \rightarrow \infty,\) we refer to the continuous compounding of interest. Use I'Hospital's Rule to show that if interest is compounded continuously, then the amount after t years is $$A=A_{0} e^{r t}$$

Ornithologists have determined that some species of birds tend to avoid flights over large bodies of water during daylight hours. It is believed that more energy is required to fly over water than over land because air generally rises over land and falls over water during the day. A bird with these tendencies is released from an island that is 5\(\mathrm { km }\) from the nearest point \(B\) on a straight shoreline, flies to a point \(C\) on the shoreline, and then flies along the shoreline to its nesting area \(D .\) Assume that the bird instinctively chooses a path that will minimize its energy expenditure. Points \(B\) and \(D\) are 13\(\mathrm { km }\) apart. (a) In general, if it takes 1.4 times as much energy to fly over water as it does over land, to what point \(C\) should the bird fly in order to minimize the total energy expended in returning to its nesting area? (b) Let \(W\) and \(L\) denote the energy (in joules) per kilometer flown over water and land, respectively. What would a large value of the ratio \(W / L\) mean in terms of the bird's flight? What would a small value mean? Determine the ratio \(W / L\) corresponding to the minimum expenditure of energy. (c) What should the value of \(W / L\) be in order for the bird to fly directly to its nesting area \(D ?\) What should the value of \(W / L\) be for the bird to fly to \(B\) and then along the shore to \(D ?\) (d) If the omithologists observe that birds of a certain species reach the shore at a point 4\(\mathrm { km }\) from \(B\) , how many times more energy does it take a bird to fly over water than over land?
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