Problem 82 - Thomas Calculus

Question

Find the limits in Exercises \(80-86\). $$\lim _{x \rightarrow-\infty}\left(\sqrt{x^{2}+3}+x\right)$$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understand the Expression

The expression we are dealing with is \( \lim_{x \rightarrow -\infty} \left( \sqrt{x^2 + 3} + x \right) \). As \( x \rightarrow -\infty \), both terms inside the parenthesis approach infinity, but at different rates. We need to find a way to simplify these terms to determine the limit.

Step 2: Simplify the Expression

Rewrite the expression inside the limit: \( \sqrt{x^2 + 3} = \sqrt{x^2(1 + \frac{3}{x^2})} = |x|\sqrt{1 + \frac{3}{x^2}} \). Since \( x \rightarrow -\infty \), \( |x| = -x \). Thus, \( \sqrt{x^2 + 3} = -x\sqrt{1 + \frac{3}{x^2}} \).

Step 3: Substitute Back into Limit

Substitute the simplified expression back into the limit: \( \lim_{x \rightarrow -\infty} \left(-x\sqrt{1 + \frac{3}{x^2}} + x\right) \). Simplify further: \( \lim_{x \rightarrow -\infty} x(-\sqrt{1 + \frac{3}{x^2}} + 1) \).

Step 4: Simplify Further and Evaluate

Notice the expression \(-\sqrt{1 + \frac{3}{x^2}} + 1\). Evaluating the limit of the expression \( \sqrt{1+\frac{3}{x^2}} \rightarrow 1 \) as \( x \rightarrow -\infty \), thus \( -\sqrt{1 + \frac{3}{x^2}} + 1 \rightarrow 0. \) Therefore, \( \lim_{x \rightarrow -\infty} x(0) = 0 \).

Step 5: Conclusion of Evaluation

Finally, the limit of the expression \( \lim_{x \rightarrow -\infty} \left( \sqrt{x^2+3} + x \right) \) evaluates to 0.

Key Concepts

Asymptotic Behavior

Asymptotic behavior refers to how a function behaves as its input approaches a certain value or infinity. In this exercise, we are concerned with the behavior of the function as \( x \) approaches \(-\infty\). Understanding asymptotic behavior allows us to predict the trend of a graph, especially for large values of \( x \).When analyzing the function \( \sqrt{x^2 + 3} + x \), we note that each term contributes differently to the function's behavior at negative infinity. For very large negative values of \( x \), the term \( x \) is crucial, as it tends to dominate due to its linear nature when compared to terms inside a square root, highlighting the importance of examining growth rates in asymptotic analysis.Overall, asymptotic behavior helps us see beyond immediate values to the important trends and tendencies of a function as \( x \) stretches out towards \(-\infty\), providing insight into its limit.

Infinity Limits

Infinity limits deal with the behavior of functions as the variable approaches positive or negative infinity. In this scenario, the challenge is to find the limit of the expression \( \lim_{x \rightarrow -\infty} ( \sqrt{x^2 + 3} + x) \).Here's the thought process:

• As \( x \to -\infty \), both \( \sqrt{x^2 + 3} \) and \( x \) individually approach infinity, but at different rates.
• To tackle this, we simplify the square root using algebraic manipulation, turning it into \(-x\sqrt{1 + \frac{3}{x^2}}\).

The simplification reveals that both \( \sqrt{x^2 + 3} \) and \( x \) move with the same asymptotic rate. We aim to analyze whether these terms cancel each other out, converge, or diverge when added. This approach unveils that the infinity limit for this scenario resolves to zero.

Square Root Simplification

Square root simplification is a crucial technique in solving limits, particularly those involving expressions like \( \sqrt{x^2 + 3} \). The process involves reshaping the expression to evaluate extreme behavior more easily.

• We rewrote \( \sqrt{x^2 + 3} \) as \( \sqrt{x^2(1 + \frac{3}{x^2})} \).
• This allowed us to express it as \(-x\sqrt{1 + \frac{3}{x^2}}\) when \( x \) heads to \(-\infty\), given \(|x| = -x \) at such values.

Simplifying complex square roots by factoring out the dominant term, \( x^2 \), makes it feasible to determine limits at infinity. By focusing on the expression \( \sqrt{1 + \frac{3}{x^2}} \), the seemingly complex behavior simplifies, showing as \( x \) becomes very large and negative, this expression approximates to 1.Knowledge of square root simplification streamlines tackling intricate limit problems, transforming them into manageable steps toward a solution.