Finding Limits of Differences When x → ±∞

Find the limits in Exercises 84–90. (Hint: Try multiplying and dividing by the conjugate.)

lim x → −∞ (√(x² + 3) + x)

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Identify the expression for which you need to find the limit: \( \lim_{x \to -\infty} (\sqrt{x^2 + 3} + x) \).

To simplify the expression, multiply and divide by the conjugate: \( \frac{(\sqrt{x^2 + 3} + x)(\sqrt{x^2 + 3} - x)}{\sqrt{x^2 + 3} - x} \).

The numerator becomes a difference of squares: \((\sqrt{x^2 + 3})^2 - x^2 = x^2 + 3 - x^2 = 3\).

Now, the expression simplifies to: \( \frac{3}{\sqrt{x^2 + 3} - x} \).

Analyze the behavior of the denominator as \( x \to -\infty \). Simplify \( \sqrt{x^2 + 3} \approx |x| \) for large \( |x| \), and consider the limit of the simplified expression.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits at Infinity

Limits at infinity involve evaluating the behavior of a function as the variable approaches positive or negative infinity. This concept is crucial for understanding how functions behave in extreme cases, often revealing horizontal asymptotes or end behavior. In the context of the given limit, we analyze how the expression behaves as x becomes very large or very small.

Conjugates in Limits

Using conjugates is a technique in calculus that simplifies expressions involving square roots. By multiplying and dividing by the conjugate, we can eliminate square roots in the numerator or denominator, making it easier to evaluate limits. This method is particularly useful when dealing with indeterminate forms that arise in limit calculations.

Square Root Functions

Square root functions, such as √(x² + 3), are essential in calculus for understanding how they behave as x approaches infinity or negative infinity. The square root of a squared term dominates the behavior of the function, allowing us to simplify the limit. Recognizing how these functions interact with linear terms is key to solving the limit problem presented.

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