Finding Limits of Differences When x → ±∞


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


lim x → ∞ (√(x + 9) − √(x + 4))

Verified step by step guidance

1. Identify the expression whose limit you need to find: \( \lim_{x \to \infty} (\sqrt{x + 9} - \sqrt{x + 4}) \).
2. To simplify the expression, multiply and divide by the conjugate: \( \frac{(\sqrt{x + 9} - \sqrt{x + 4})(\sqrt{x + 9} + \sqrt{x + 4})}{\sqrt{x + 9} + \sqrt{x + 4}} \).
3. The numerator becomes a difference of squares: \((x + 9) - (x + 4) = 5\).
4. The expression simplifies to \( \frac{5}{\sqrt{x + 9} + \sqrt{x + 4}} \).
5. As \( x \to \infty \), both \( \sqrt{x + 9} \) and \( \sqrt{x + 4} \) approach \( \sqrt{x} \), so the denominator approaches \( 2\sqrt{x} \). Thus, the limit becomes \( \lim_{x \to \infty} \frac{5}{2\sqrt{x}} \), which approaches 0.

Key Concepts

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In this context, we are interested in the behavior of the function as x approaches infinity, which helps us understand the end behavior of the function and how it simplifies under such conditions.

Conjugates

The conjugate of a binomial expression is formed by changing the sign between two terms. In the context of limits, multiplying by the conjugate can help eliminate square roots or simplify expressions, making it easier to evaluate the limit. This technique is particularly useful when dealing with differences of square roots.

Rationalization

Rationalization is a method used to eliminate radicals from the denominator or simplify expressions involving square roots. By multiplying the numerator and denominator by the conjugate, we can transform the expression into a more manageable form, allowing for easier calculation of limits, especially as x approaches infinity.