Question

Use analytical methods to evaluate the following limits. $$\lim _{x \rightarrow \infty}(\sqrt{x-2}-\sqrt{x-4})$$

Short answer

Answer: The limit of the function as $x$ approaches infinity is 0.

Step-by-step solution

Rationalize the expression

To rationalize the expression, multiply the numerator and denominator of the function by the conjugate of this numerator: \((\sqrt{x-2}+\sqrt{x-4})\). This helps us to eliminate the square root in the numerator: $$\frac{\sqrt{x-2}-\sqrt{x-4}}{1} \times \frac{\sqrt{x-2}+\sqrt{x-4}}{\sqrt{x-2}+\sqrt{x-4}} = \frac{(x-2)-(x-4)}{\sqrt{x-2}+\sqrt{x-4}}$$ Step 2: Simplify the expression

Simplify the expression

Now simplify the numerator and combine terms: $$\lim _{x \rightarrow \infty} \frac{2}{\sqrt{x-2}+\sqrt{x-4}}$$ Step 3: Apply the limit

Apply the limit

In this stage, we need to find out what happens to our expression when x approaches infinity: $$\lim _{x \rightarrow \infty} \frac{2}{\sqrt{x-2}+\sqrt{x-4}}$$ As x approaches infinity, both square roots in the denominator approach infinity as well. However, the sum of the two square roots also approaches infinity, so we can rewrite the expression as: $$\lim _{x \rightarrow \infty} \frac{2}{\infty} = 0$$ The limit of the expression, as x approaches infinity, is 0.

Key concepts

Analytical Methods in Calculus

Calculus is a branch of mathematics that deals with the study of change and motion. Within calculus, we often encounter challenges that require us to evaluate limits, derivatives, and integrals. One of the most powerful tools we have for understanding the behavior of functions is limit evaluation. Analytical methods in calculus involve mathematical techniques used to manipulate functions and expressions to determine limits analytically rather than numerically or graphically.

For example, evaluating the limit of a function as it approaches infinity is a common task. In the provided exercise, analytical methods come into play by rationalizing expressions — a technique crucial for simplifying square roots — and then strategically manipulating the expression to reveal its behavior at the limits. By understanding and applying these methods, one can make sophisticated predictions about the function's long-term behavior, which is particularly useful in fields such as engineering and physics.

Rationalizing Expressions

Rationalizing expressions is a technique used to eliminate radicals from denominators or numerators. The overall goal is to simplify expressions, particularly when involving square roots or other irrational numbers. This is done by multiplying by a 'conjugate', which is a binomial formed by changing the sign between two terms.

In the context of our example, we have two square roots in the numerator, which poses a challenge for limit evaluation. By multiplying by the conjugate, we simplify the expression and make it easier to handle, especially as we look towards the limit at infinity. Rationalizing can also be essential when dealing with more complex problems such as evaluating integrals or solving equations, where the presence of square roots can complicate the mathematical process.

Limits at Infinity

In calculus, 'limits at infinity' is a concept that encapsulates the behavior of a function as the input grows larger and larger, approaching infinity. This is crucial for understanding a function's end behavior, which can tell us about its asymptotic trends and horizontal asymptotes.

It is important to recognize that as numbers become excessively large, differences in constant terms become negligible. For example, in the given exercise, both \(\sqrt{x-2}\) and \(\sqrt{x-4}\) become increasingly large, and their difference insignificant, approaching infinity. Hence, we can consider their rates of growth and simplify our analysis by focusing on their dominant behavior. Limits at infinity often result in insights about the existence of horizontal asymptotes, which can be pivotal in sketching function graphs and predicting long-term trends.

Square Root Simplification

Simplifying square root expressions is a common step in solving calculus problems. It involves the process of rewriting a square root in a simpler form, sometimes by extracting factors that are perfect squares or using rationalization techniques as seen in the exercise.

When we simplify square roots within the context of limits, we're often preparing the expression for other operations, such as differentiation or further limit evaluations. In the exercise provided, simplification occurs after rationalization, which ultimately helps us discern that the limit as x approaches infinity is zero. Simplification of square roots can also be critical in other areas of calculus, such as when finding derivatives or antiderivatives, because it might reduce the function to a form that's more easily differentiated or integrated.