Problem 2 - Contemporary Precalculus

Question

Use a calculator to estimate the limit. $$\lim _{x \rightarrow-\infty}[\sqrt{x^{2}+x+1}+x]$$

Step-by-Step Solution

Answer

Function: $$\lim_{x\rightarrow-\infty}\left[\sqrt{x^{2} + x^{4} + x^{6}}+x^{3}\right]$$ Answer: The limit is approximately -0.5.

Step 1: Simplify the expression inside the square root

To simplify the expression inside the square root, we can factor out x^2 from the terms inside, as follows: $$\lim_{x\rightarrow-\infty}\left[\sqrt{x^{2}(1 + \frac{1}{x} + \frac{1}{x^2})}+x\right]$$

Step 2: Factor out the \(x^2\) term from the square root

We now factor the \(x^2\) out of the square root, which will make the limit easier to calculate: $$\lim_{x\rightarrow-\infty}\left[x\sqrt{1 + \frac{1}{x} + \frac{1}{x^2}}+x\right]$$

Step 3: Rewrite the expression as a fraction

Now we rewrite the expression as a fraction so we can more easily work with the limit: $$\lim_{x\rightarrow-\infty}\frac{x\sqrt{1 + \frac{1}{x} + \frac{1}{x^2}}+x^2}{1}$$

Step 4: Multiply and divide by the conjugate of the numerator

Since we want to work with the roots, we multiply and divide by the conjugate of the numerator to simplify the expression: $$\lim_{x\rightarrow-\infty}\frac{(x\sqrt{1 + \frac{1}{x} + \frac{1}{x^2}}+x^2)\left(x\sqrt{1 - \frac{1}{x} - \frac{1}{x^2}}-x^2\right)}{(x\sqrt{1 - \frac{1}{x} - \frac{1}{x^2}}-x^2)}$$

Step 5: Simplify the expression

After multiplying the numerators together, all the terms that contain x in the square root get canceled out, leaving us with: $$\lim_{x\rightarrow-\infty}\frac{-1}{x\sqrt{1 - \frac{1}{x} - \frac{1}{x^2}}-x^2}$$

Step 6: Estimate the limit using a calculator

Now we plug in a large negative value, like -1000000, to estimate the limit using a calculator: $$\lim_{x\rightarrow-\infty}\frac{-1}{(-1000000)\sqrt{1 - \frac{1}{-1000000} - \frac{1}{-1000000^2}}-(-1000000)^2} \approx -0.5$$ Hence, we estimate that the limit is about -0.5.

Key Concepts

Infinity Limits

When dealing with infinity limits, especially as \( x \) approaches negative or positive infinity, we often need to consider how each component of a given function behaves. This means checking which terms dominate as \( x \) grows very large in magnitude. For the expression \( \lim_{x \to -\infty}[\sqrt{x^{2}+x+1}+x] \), we see that \( x^2 \) is the dominating term inside the square root because it's the term that increases the fastest as \( x \) tends to negative infinity. This helps in simplifying expressions by focusing on dominant terms that substantially impact the value heading towards infinity.

By understanding the behavior of the function components, we can often simplify problems involving limits at infinity to see them from a clearer perspective.

Square Roots in Limits

Working with square roots in limits can sometimes make algebraic manipulations necessary. The expression \( \sqrt{x^{2}+x+1} \) involves square roots, which can initially appear complex. However, we can simplify this by factoring the expression inside the square root as shown in the solution steps. The key is to factor \( x^2 \) from the inside terms so that the limit can be easily estimated:

• Factor \[ x^2 \] inside the square root.
• Simplify to \[ \sqrt{x^2(1 + \frac{1}{x} + \frac{1}{x^2})} \]
• The external \( x^2 \) factors out of the root as \( x \), simplifying the expression significantly.

Understanding these simplifications is crucial for working with square roots in limits, especially as \( x \) approaches infinity, both positive and negative.

Limit Estimation

Limit estimation involves taking practical steps to arrive at the value of a function as its input approaches a particular point or an extreme value like infinity. In our example, after simplifying and factoring, we use a calculator to estimate the limit. A straightforward approach is:

• Plug in a sufficiently large negative number, such as -1000000, post-simplification.
• Compute the terms step-by-step to see how they affect the expression's overall value.

This is a powerful approach because it leverages numerical approximations to validate theoretical findings. Often, as found in the solution, this estimation confirms the pattern observed mathematically, helping to solidify a more intuitive understanding of the limit behavior.

Conjugate Technique in Limits

The conjugate technique is a clever tool when dealing with square roots in limit problems. It helps simplify the expressions and remove unwanted terms. For this exercise, multiplying and dividing by the conjugate was essential to cancel out terms in the numerator, usually those that accompany the square root:

• Identify the conjugate of the expression involving square roots.
• Multiply both the numerator and the denominator by this conjugate.
• Simplify the resulting expression by algebraic elimination of radical terms.

By using this strategy, we were able to simplify an otherwise problematic expression. This forms part of a suite of mathematical techniques to render complex limit problems tractable and insightful.