Problem 21
Question
Estimating Limits Numerically and Graphically Use a table of values to estimate the limit. Then use a graphing device to confirm your result graphically. $$\lim _{x \rightarrow \infty} \frac{x^{5}}{e^{x}}$$
Step-by-Step Solution
Verified Answer
The limit \( \lim_{x \to \infty} \frac{x^5}{e^x} = 0 \).
1Step 1: Set Up a Table of Values
To estimate \( \lim_{x \to \infty} \frac{x^5}{e^x} \), we can substitute increasingly large values for \( x \). Let's choose values such as 10, 100, and 1000 to see how the function behaves. Calculate the function \( f(x) = \frac{x^5}{e^x} \) for each of these values.
2Step 2: Compute Values for Large x
Calculate the values: - For \( x = 10 \), \( \frac{10^5}{e^{10}} \). - For \( x = 100 \), \( \frac{100^5}{e^{100}} \). - For \( x = 1000 \), \( \frac{1000^5}{e^{1000}} \).Observe that these values get exceedingly close to zero.
3Step 3: Analyze the Numerical Trend
From step 2, notice that as \( x \) increases, \( \frac{x^5}{e^x} \) becomes very small and approaches zero. This suggests that the limit is zero as \( x \to \infty \).
4Step 4: Graph the Function
Use a graphing calculator or software to plot \( f(x) = \frac{x^5}{e^x} \). The graph should show that as \( x \) increases, the value of \( f(x) \) approaches zero, confirming the estimated numerical limit.
5Step 5: Conclusion: Estimate and Confirm Limit
Both numerical calculation and graphical representation show that \( \lim_{x \to \infty} \frac{x^5}{e^x} = 0 \). The function decreases and approaches zero as \( x \to \infty \).
Key Concepts
Numerical EstimationGraphical ConfirmationLimits at InfinityExponential Growth vs Polynomial Growth
Numerical Estimation
Numerical estimation is a useful technique when trying to determine the limit of a function as a variable approaches a particular value. In the case of \( \lim_{x \rightarrow \infty} \frac{x^5}{e^x} \), the process involves substituting progressively larger values for \( x \) to observe how the function behaves. For instance, after computing values such as \( x = 10, 100, \) and \( 1000 \), you'll notice that the function diminishes closer to zero each time.
To do this, calculate \( f(x) = \frac{x^5}{e^x} \) for each chosen \( x \):
This numerical approach gives a tangible sense of how the function behaves as \( x \) arrives at infinity, helping to confirm or refute the anticipated outcome through computational evidence.
To do this, calculate \( f(x) = \frac{x^5}{e^x} \) for each chosen \( x \):
- For \( x = 10 \), \( f(x) = \frac{10^5}{e^{10}} \).
- For \( x = 100 \), \( f(x) = \frac{100^5}{e^{100}} \).
- For \( x = 1000 \), \( f(x) = \frac{1000^5}{e^{1000}} \).
This numerical approach gives a tangible sense of how the function behaves as \( x \) arrives at infinity, helping to confirm or refute the anticipated outcome through computational evidence.
Graphical Confirmation
Graphical confirmation is a visual method to verify the conclusions derived from numerical estimation. By graphing the function \( f(x) = \frac{x^5}{e^x} \), you provide a pictorial representation that complements numerical results.
Using a graphing tool, such as a calculator or computer software, you can plot the function over a suitable range of \( x \). As you observe the graph, pay attention to the trend of \( f(x) \) as \( x \) becomes very large.
The graph should depict that the values of \( f(x) \) decrease swiftly and approach zero as \( x \) increases. This visual evidence works alongside the calculated values to provide a comprehensive understanding, confirming that \( \lim_{x \rightarrow \infty} \frac{x^5}{e^x} = 0 \).
Graphical analysis not only confirms numeric results but also offers a deeper insight into the behavior of the function over an infinite domain.
Using a graphing tool, such as a calculator or computer software, you can plot the function over a suitable range of \( x \). As you observe the graph, pay attention to the trend of \( f(x) \) as \( x \) becomes very large.
The graph should depict that the values of \( f(x) \) decrease swiftly and approach zero as \( x \) increases. This visual evidence works alongside the calculated values to provide a comprehensive understanding, confirming that \( \lim_{x \rightarrow \infty} \frac{x^5}{e^x} = 0 \).
Graphical analysis not only confirms numeric results but also offers a deeper insight into the behavior of the function over an infinite domain.
Limits at Infinity
Limits at infinity involve evaluating the behavior of a function as its variable approaches infinity. It's crucial in understanding the end behavior of functions, particularly when comparing growth rates of different expressions.
In the example \( \lim_{x \rightarrow \infty} \frac{x^5}{e^x} \), we're interested in how the polynomial numerator \( x^5 \) compares to the exponential denominator \( e^x \) as \( x \) increases.
Since exponential functions grow much faster than polynomial functions, the exponential term \( e^x \) in the denominator becomes exceedingly large much quicker than the growth of \( x^5 \). Consequently, the overall value of \( \frac{x^5}{e^x} \) approaches zero.
Understanding limits at infinity helps predict the behavior of complex functions and supports long-term predictions about their growth or decay.
In the example \( \lim_{x \rightarrow \infty} \frac{x^5}{e^x} \), we're interested in how the polynomial numerator \( x^5 \) compares to the exponential denominator \( e^x \) as \( x \) increases.
Since exponential functions grow much faster than polynomial functions, the exponential term \( e^x \) in the denominator becomes exceedingly large much quicker than the growth of \( x^5 \). Consequently, the overall value of \( \frac{x^5}{e^x} \) approaches zero.
Understanding limits at infinity helps predict the behavior of complex functions and supports long-term predictions about their growth or decay.
Exponential Growth vs Polynomial Growth
Exponential growth and polynomial growth are two types of growth patterns often compared in calculus. They become especially crucial when evaluating limits at infinity, as seen in the exercise with \( f(x) = \frac{x^5}{e^x} \).
Exponential growth is characterized by a constant base raised to a variable exponent, like \( e^x \). This function grows much faster than polynomial expressions, especially for large \( x \).
Exponential growth is characterized by a constant base raised to a variable exponent, like \( e^x \). This function grows much faster than polynomial expressions, especially for large \( x \).
- For example, if you compare \( x^5 \) (a polynomial) with \( e^x \) (an exponential), \( e^x \) will always outgrow \( x^5 \) as \( x \) increases.
- This is why in \( \frac{x^5}{e^x} \), the denominator causes the function to shrink towards zero as \( x \) gets larger.
Other exercises in this chapter
Problem 21
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