Problem 11 - Calculus: Early Transcendentals

Question

Guess the value of the limit $$ \lim_{x \to \infty} \frac{x^2}{2^x} $$ by evaluating the function \( f(x) = x^2/2^x \) for \( x \) = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 50, and 100. Then use a graph of \( f \) to support your guess.

Step-by-Step Solution

Answer

The limit is 0 as \( x \to \infty \).

Step 1: Evaluate the Function at Given Points

Calculate the value of the function \( f(x) = \frac{x^2}{2^x} \) for each specified \( x \): 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 50, and 100.- For \( x = 0 \), \( f(0) = \frac{0^2}{2^0} = 0 \)- For \( x = 1 \), \( f(1) = \frac{1^2}{2^1} = 0.5 \)- For \( x = 2 \), \( f(2) = \frac{2^2}{2^2} = 1 \)- For \( x = 3 \), \( f(3) = \frac{3^2}{2^3} = \frac{9}{8} = 1.125 \)- For \( x = 4 \), \( f(4) = \frac{4^2}{2^4} = \frac{16}{16} = 1 \)- For \( x = 5 \), \( f(5) = \frac{5^2}{2^5} = \frac{25}{32} \approx 0.78125 \)- For \( x = 6 \), \( f(6) = \frac{6^2}{2^6} = \frac{36}{64} = 0.5625 \)- For \( x = 7 \), \( f(7) = \frac{7^2}{2^7} = \frac{49}{128} \approx 0.38281 \)- For \( x = 8 \), \( f(8) = \frac{8^2}{2^8} = \frac{64}{256} = 0.25 \)- For \( x = 9 \), \( f(9) = \frac{9^2}{2^9} = \frac{81}{512} \approx 0.15820 \)- For \( x = 10 \), \( f(10) = \frac{10^2}{2^{10}} = \frac{100}{1024} \approx 0.09766 \)- For \( x = 20 \), \( f(20) = \frac{20^2}{2^{20}} = \frac{400}{1048576} \approx 0.000381 \)- For \( x = 50 \), \( f(50) = \frac{50^2}{2^{50}} = \frac{2500}{1125899906842624} \approx 2.22 \times 10^{-12} \)- For \( x = 100 \), \( f(100) = \frac{100^2}{2^{100}} = \frac{10000}{1.27 \times 10^{30}} \approx 7.88 \times 10^{-28} \)

Step 2: Analyze the Results

Notice that as \( x \) increases, the value of \( f(x) = \frac{x^2}{2^x} \) decreases. Initially, the values are relatively large (like 0.5 for \( x = 1 \) and 1 for \( x = 2 \)), but they rapidly decrease to extremely small numbers as \( x \) becomes larger, suggesting a tendency towards zero.

Step 3: Graph the Function

Plot the function \( f(x) = \frac{x^2}{2^x} \) using a graphing tool or software. The graph will show how the function behaves as \( x \) approaches infinity. The plot should confirm the trend seen in the calculated values: as \( x \) increases, \( f(x) \) approaches zero.

Step 4: Conclude about the Limit

Based on the calculated values and the plot, you can conclude that as \( x \) approaches infinity, the limit of \( \frac{x^2}{2^x} \) tends to zero. This conclusion is supported by both the numerical evaluation and the graphical representation of the function.

Key Concepts

Infinite Limits

When we talk about limits in calculus, we're discussing how a function behaves as its input (usually denoted as \( x \)) approaches a certain value. An infinite limit describes what happens as \( x \) either grows very large (approaches infinity) or very small (approaches negative infinity). This topic is vital for understanding the behavior of functions at the extremes.
In our exercise, where we analyze the limit of \( \lim_{x \to \infty} \frac{x^2}{2^x} \), we evaluate what happens when \( x \) gets very large. By examining values at various points (e.g., \( x = 10, 20, 50, 100 \)), we notice that as \( x \) increases, the function \( \frac{x^2}{2^x} \) gets smaller and smaller, approaching zero. This is because in the expression, although \( x^2 \) grows as \( x \) increases, \( 2^x \) grows much faster, leading the whole fraction to shrink.
In summary, infinite limits help us determine how functions behave at the metaphorical edges. In this example, despite both the numerator and denominator increasing, the exponential term \( 2^x \)'s rapid growth dominates, driving the limit towards zero.

Function Analysis

Function analysis involves breaking down a mathematical expression to understand its characteristics, behavior, and potential limits.
For the function \( f(x) = \frac{x^2}{2^x} \), we dissect it to see how different parts of the function contribute to its overall behavior. The numerator, \( x^2 \), grows quadratically, meaning it increases continually as \( x \) grows. However, the denominator, which is an exponential function (\( 2^x \)), increases much faster than any polynomial, including \( x^2 \). This difference is crucial for determining the limit as \( x \) approaches infinity.
By computing the function at discrete points, such as \( x = 0, 1, 2, ..., 100 \), we observed that \( f(x) \) starts at a higher value and progressively decreases. The calculations in the step-by-step solution show a consistent trend where \( f(x) \) approaches zero for large \( x \). Understanding the conflict between the growth rates of the polynomial and exponential parts is key to this analysis.

Graphical Representation of Functions

Graphs serve as an invaluable tool in visualizing functions and their limits. A graphical representation can provide insights that might not be immediately obvious from formulas and calculations alone.
In this scenario, plotting \( f(x) = \frac{x^2}{2^x} \) helps us see the trend of the function more clearly. As \( x \) extends towards infinity, if we plot the function using software or a graphing calculator, we visually confirm that the function's values decrease towards zero. The steep decline after a certain point shows clearly on the graph, illustrating the rapid growth of \( 2^x \) overtaking that of \( x^2 \).
Graphing also helps identify key points such as where the function peaks or flattens out, reinforcing our analytical findings. In our exercise, the graph bolsters the conclusion that the limit of \( \frac{x^2}{2^x} \) as \( x \) approaches infinity is indeed zero. This graphical insight complements computational analyses, enabling a comprehensive understanding.