Problem 45 - Calculus: Early Transcendentals

Question

(a) Estimate the value of $$ \lim_{x \to -\infty} \left( \sqrt{x^2 + x + 1} + x \right) $$ by graphing the function \( f(x) = \sqrt{x^2 + x + 1} + x \). (b) Use a table of values of \( f(x) \) to guess the value of the limit. (c) Prove that your guess is correct.

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Write the Function

The given function is \( f(x) = \sqrt{x^2 + x + 1} + x \). We need to analyze this function as \( x \) approaches negative infinity.

Step 2: Graph the Function

Graph \( f(x) = \sqrt{x^2 + x + 1} + x \) using graphing software or a graphing calculator. Look at the behavior of the function as \( x \to -\infty \). As \( x \) becomes very negative, notice that the value of \( f(x) \) appears to approach 0.

Step 3: Create a Table of Values

Calculate \( f(x) \) for values like \( x = -1, -10, -100, -1000 \), etc. For example: \( f(-1) \approx 1.732 \), \( f(-10) \approx 0.05 \), \( f(-100) \approx 0.005 \), \( f(-1000) \approx 0.0005 \). The outputs are decreasing and approach zero as \( x \) becomes more negative.

Step 4: Analyze the Expression

Consider \( f(x) = \sqrt{x^2 + x + 1} + x \) and simplify. As \( x \to -\infty \), we approximate \( \sqrt{x^2 + x + 1} \approx |x| \), which is \( -x \) for negative \( x \). Substitute this approximation in to get \( f(x) \approx -x + x = 0 \).

Step 5: Conclusion of Limit

Both empirical evidence from graphing and logical analysis confirm that \( \lim\limits_{x \to -\infty} (\sqrt{x^2 + x + 1} + x) = 0 \). Thus, the limit is 0.

Key Concepts

Graphical estimation of limits

To estimate a limit graphically, we start by plotting the function. In our case, the function we were given is \( f(x) = \sqrt{x^2 + x + 1} + x \).By using graphing software or a calculator, we can plot this function over a range of negative values for \( x \). Once we have the graph in front of us, we're interested in observing the behavior as \( x \) approaches negative infinity.

This means we're looking at what happens to \( f(x) \) when \( x \) gets very, very negative. From such a graph, if you notice that the curve is leveling off towards a particular value, you can estimate the limit. In our specific problem, it became clear from the graph that as \( x \) trends towards the left, \( f(x) \) approaches 0. This estimation suggests that the limit is 0, but it's merely a sketch and needs further verification.

Table of values

After visually estimating the limit using a graph, it's beneficial to use calculated numerical values to support our hypothesis. We do this by creating a table of values. For this function, we calculated \( f(x) \) for several negative values of \( x \), such as \( x = -1, -10, -100, -1000 \).

With these calculated values:

• \( f(-1) \approx 1.732 \)
• \( f(-10) \approx 0.05 \)
• \( f(-100) \approx 0.005 \)
• \( f(-1000) \approx 0.0005 \)

You'll notice that as \( x \) becomes more negative, the function value gets closer to 0. This table helps confirm our graphical observation, providing a quantitative look at the trend: as \( x \) approaches negative infinity, \( f(x) \) indeed seems to approach 0.

Proof of limits

Once we've graphically and numerically estimated the limit, it's time to confirm with a formal proof. We can revisit the algebraic expression of \( f(x) = \sqrt{x^2 + x + 1} + x \). For very negative \( x \), the term \( \sqrt{x^2 + x + 1} \) needs simplification to better understand its behavior as \( x \to -\infty \).

For large absolute values of \( x \), the dominant term inside the square root is \( x^2 \), leading to the approximation \( \sqrt{x^2 + x + 1} \approx |x| \). Because we're considering \( x \) being negative, \( |x| = -x \), simplifying the expression to form:\[ f(x) \approx -x + x = 0 \].This simplification aligns perfectly with our prior graphical and numerical findings.

Completing the proof confirms that both our estimation methods and logical reasoning hold true, ensuring that \( \lim_{x \to -\infty} (\sqrt{x^2 + x + 1} + x) = 0 \). This is a solid example of how visualization, numerical evidence, and mathematical rigor work together to solidify our understanding of limits.