Question

In Exercises 2.4.2-2.4.40, find the indicated limits. $$ \lim _{x \rightarrow \infty}(\sqrt{x+1}-\sqrt{x}) $$

Short answer

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

Step-by-step solution

Write the given expression.

The given expression is: $$ \lim_{x \rightarrow \infty}(\sqrt{x+1}-\sqrt{x}) $$

Rationalize the expression.

In order to simplify the expression, we can rationalize it. To do so, multiply the numerator and denominator by the conjugate of the given expression: $$ \lim_{x \rightarrow \infty}\frac{(\sqrt{x+1}-\sqrt{x})(\sqrt{x+1}+\sqrt{x})}{(\sqrt{x+1}+\sqrt{x})} $$

Simplify the expression.

Now, we can simplify the expression by performing the multiplication in the numerator: $$ \lim_{x \rightarrow \infty}\frac{(x+1)-(\sqrt{x+1}\sqrt{x}+\sqrt{x}\sqrt{x+1})+x}{(\sqrt{x+1}+\sqrt{x})} $$ Combine the terms and further simplify the expression: $$ \lim_{x \rightarrow \infty}\frac{1-\sqrt{x(x+1)}(1+1)}{(\sqrt{x+1}+\sqrt{x})} $$

Find the limit as x approaches infinity.

Divide every term in the expression by x inside the square root to get: $$ \lim_{x \rightarrow \infty}\frac{\frac{1}{x}}{\sqrt{\frac{x+1}{x}}+\sqrt{1}} $$

Apply the limit to the simplified expression and solve.

Now, we can apply the limit and solve the expression: $$ \lim_{x \rightarrow \infty}\frac{\frac{1}{x}}{\sqrt{1+\frac{1}{x}}+\sqrt{1}} = \frac{0}{\sqrt{1+0}+\sqrt{1}} = \boxed{0} $$ Hence, the limit of the given expression as x approaches infinity is 0.

Key concepts

Rationalizing Expressions

Rationalizing expressions is a technique used to simplify complex terms, especially those involving square roots or other irrational numbers. This is particularly useful in calculus when trying to evaluate limits or simplify expressions for easier computation. To rationalize an expression like \( \sqrt{x+1}-\sqrt{x} \), we multiply the expression by its conjugate. The conjugate of \( \sqrt{x+1}-\sqrt{x} \) is \( \sqrt{x+1}+\sqrt{x} \).

By multiplying the original expression by its conjugate, we eliminate the square roots in the numerator, simplifying the calculation. The process involves these steps:

• Multiply both the numerator and the denominator by the conjugate.
• Simplify the expression, convert terms, and cancel where possible.

Rationalizing is a powerful tool not just for limits involving infinity, but also simplifying algebraic fractions in general, making solutions clearer and more straightforward.

Infinite Limits

Infinite limits refer to the behavior of a function as the input variable approaches infinity or negative infinity. Such limits help determine the asymptotic behavior of functions and are essential in calculus to understand trends and predict patterns.

In the case of \( \lim_{x \rightarrow \infty}(\sqrt{x+1}-\sqrt{x}) \), we are exploring what happens to the expression as \( x \) grows larger. To analyze infinite limits, it's often beneficial to simplify the expression first, like rationalizing or dividing terms by high powers of \( x \).

The calculation here involves examining terms such as \( \frac{1}{x} \) where each term tends toward 0 as \( x \) approaches infinity. The resulting expression \( \frac{0}{\text{finite number}} \) confirms the behavior of the whole function towards being akin to approaching zero.

Square Roots in Limits

Square roots can introduce complexity into calculus problems, especially when they appear in limit problems. Understanding how square roots behave when approaching infinity or other critical values is pivotal.

In limits, square roots often require a companion technique, such as rationalization, for simplification. Consider the expression \( \sqrt{x+1}-\sqrt{x} \). Individually, each square root would approach a unique value as \( x \to \infty \), but together in subtraction, they form a unique interaction. By rationalizing, we simplify their appearance in calculations.

The behavior of square roots under limits often also involves comparing terms inside and outside the root. Here, breaking down \( \sqrt{x+1} \) as \( \sqrt{x(1+\frac{1}{x})} \) aids in understanding how additional terms diminish and simplify the entire expression, facilitating the evaluation of the limit.