Problem 7 - Calculus for the Life Sciences: A Modeling Approach Volume II

Question

Exercise 14.5.7 L'Hôspital's Rule can be cascaded. To show that \(\lim _{t \rightarrow \infty} \frac{e^{t}}{t^{2}}=\infty, \quad\) observe first that \(\quad \lim _{t \rightarrow \infty} \frac{e^{t}}{t}=\infty .\) Let \(F(t)=e^{t}\) and \(G(t)=t^{2}\). Then \(F^{\prime}(t)=e^{t}\) and \(G^{\prime}(t)=2 t\) and \(G^{\prime}(t)>0\) for \(t>1\). Furthermore, $$ \begin{aligned} \lim _{t \rightarrow \infty} \frac{F^{\prime}(t)}{G^{\prime}(t)}=& \lim _{t \rightarrow \infty} \frac{e^{t}}{2 t}=\infty \\ \text { Therefore, } & \lim _{t \rightarrow \infty} \frac{F(t)}{G(t)}=\lim _{t \rightarrow \infty} \frac{e^{t}}{t^{2}}=\infty \end{aligned} $$ Show that a. \(\lim _{t \rightarrow \infty} \frac{e^{t}}{t^{3}}=\infty\). b. \(\quad \lim _{t \rightarrow \infty} \frac{e^{t}}{t^{4}}=\infty\). c. \(\lim _{t \rightarrow \infty} \frac{e^{t}}{t^{n}}=\infty \quad n=5,6,7, \cdots\).

Step-by-Step Solution

Answer

All given limits tend to infinity for \( n \geq 3 \) due to exponential growth.

Step 1: Define the functions for part a

For part (a), let's consider the functions \( F(t) = e^t \) and \( G(t) = t^3 \). We know that to determine the limit \( \lim _{t \rightarrow \infty} \frac{e^{t}}{t^{3}} \), we can use L'Hôpital's Rule.

Step 2: Compute the derivatives

The derivative of \( F(t) \) is \( F'(t) = e^t \). The derivative of \( G(t) \) is \( G'(t) = 3t^2 \).

Step 3: Apply L'Hôpital's Rule for part a

Using L'Hôpital's Rule, we replace the original functions with their derivatives:\[\lim _{t \rightarrow \infty} \frac{e^t}{3t^2} = \infty\]This is because \( e^t \) grows exponentially while \( 3t^2 \) grows polynomially, ensuring their ratio tends to infinity.

Step 4: Define the functions for part b

For part (b), we consider \( F(t) = e^t \) and \( G(t) = t^4 \).

Step 5: Compute the derivatives for part b

The derivative of \( F(t) \) remains \( e^t \), and the derivative of \( G(t) \) is \( G'(t) = 4t^3 \).

Step 6: Apply L'Hôpital's Rule for part b

Applying L'Hôpital's Rule for part (b), we have:\[\lim _{t \rightarrow \infty} \frac{e^t}{4t^3} = \infty\]Again, \( e^t \) grows faster than the polynomial \( 4t^3 \), so their ratio tends to infinity.

Step 7: Generalize to part c

For part (c), we want \( F(t) = e^t \) and \( G(t) = t^n \) where \( n \geq 5 \).

Step 8: Compute the derivatives for part c

The derivative of \( F(t) \) is still \( e^t \), and the derivative of \( G(t) \) is \( G'(t) = nt^{n-1} \).

Step 9: Apply L'Hôpital's Rule for part c

Using L'Hôpital's Rule repeatedly, differentiate \( n-1 \) times until the denominator becomes a constant. Notice that even for large \( n \), \( e^t \) grows infinitely faster than any polynomial. Thus,\[\lim _{t \rightarrow \infty} \frac{e^t}{nt^{n-1}} = \infty\]This ensures \( \forall n \geq 5, \lim _{t \rightarrow \infty} \frac{e^t}{t^n} = \infty \).

Step 10: Conclusion

The exponential function \( e^t \) will always grow faster than any polynomial function \( t^n \) for \( n \geq 3 \). Therefore, \( \lim _{t \rightarrow \infty} \frac{e^t}{t^n} = \infty \) for any \( n \geq 3 \).

Key Concepts

Exponential Growth

In the world of mathematics, exponential growth refers to a situation where the quantity increases at a rate proportional to its current value. The function \( e^t \) is a classic example of exponential growth. It is often used to describe growth patterns in populations, compounds in finance yielding interest, and more.
What makes exponential growth particularly intriguing is that it starts off slow but quickly ascends into a steep climb. Here are some key characteristics of exponential growth:

• Starts slowly, increases sharply: Initially, the changes might seem negligible, but over time the values skyrocket.
• Infinite growth: As time \( t \) approaches infinity, \( e^t \) heads towards infinity at an accelerating pace.
• Prevalent in natural phenomena: This type of growth is common in nature, seen in things like bacteria replication or radioactive decay when inverted.

When comparing exponential growth to polynomial growth, we see that no matter the degree of the polynomial, \( e^t \) will eventually surpass it. This is a crucial observation while handling limits involving \( e^t \) and polynomial expressions.

Polynomial Functions

Polynomial functions are mathematical expressions consisting of variables, constants, and exponents combined using addition, subtraction, multiplication, and non-negative integer exponents. Common forms include quadratic \((t^2)\), cubic \((t^3)\), and higher-degree polynomials.
Here's what you should know about polynomial functions:

• Defined by their degree: The degree of a polynomial is the highest exponent of the variable in its expression, indicating how complex the function is.
• Fixed growth rate: Unlike exponential growth, polynomial functions increase steadily but their rate of increase stays regular.
• Limiting behavior: As \( t \) increases, higher-degree terms dominate, meaning they drive the polynomial's growth.

For instance, the cubic polynomial \( t^3 \) has a rate of growth determined by \( t^2 \) and \( t \) as well, but \( t^3 \) is in control at large \( t \). This becomes significant when analyzing limits, as this steady growth contrasts sharply with the fast rise of exponential functions.

Limits

In calculus, the concept of limits is central to understanding how functions behave as they approach a certain point or infinity. Limits help us understand the trends of functions and their long-term behavior.
Some core aspects of limits include:

• Predictive power: Limits allow predictions about function values as they get closer to a particular input, even when the function isn't explicitly defined at that point.
• Tool for problem-solving: Limits are crucial for techniques like L'Hôpital's Rule, which helps evaluate indeterminate forms such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).
• Fundamental in continuity: Limits help determine if a function is continuous at a point, forming the backbone of calculus.

In the context of this exercise, limits help us compare the long-term behavior of \( e^t \) versus \( t^n \). As \( t \) approaches infinity, L'Hôpital's Rule is used to demonstrate that the exponential function \( e^t \) will continue to outgrow polynomial expressions \( t^n \), regardless of \( n \). Understanding limits is vital for making these comparisons and evaluating the behavior of functions at infinity.