Question

. Evaluate \(\lim _{x \rightarrow 0}(\sqrt{x+2}-\sqrt{2}) / x .\) Hint: Rationalize the numerator by multiplying the numerator and denominator by \(\sqrt{x+2}+\sqrt{2}\)

Short answer

The limit evaluates to \(\frac{\sqrt{2}}{4}\).

Step-by-step solution

Identify the Problem Type

The problem requires you to find a limit as a variable approaches a given value. In this case, the limit is \(\lim_{x \to 0} \frac{\sqrt{x+2} - \sqrt{2}}{x}\). The mention of multiplying with the conjugate suggests this problem will need rationalization.

Rewrite the Expression

Identify and rewrite the expression so that rationalization is possible. You have \(\sqrt{x+2} - \sqrt{2}\), which can be paired with its conjugate \(\sqrt{x+2} + \sqrt{2}\) to form a difference of squares.

Multiply by the Conjugate

Multiply both the numerator and denominator by \(\sqrt{x+2} + \sqrt{2}\). This gives:\[ \lim_{x \to 0} \frac{(\sqrt{x+2} - \sqrt{2})(\sqrt{x+2} + \sqrt{2})}{x(\sqrt{x+2} + \sqrt{2})} \]

Simplify Using the Difference of Squares

The numerator simplifies to a difference of squares: \((\sqrt{x+2})^2 - (\sqrt{2})^2 = (x+2) - 2 = x\). The expression now is:\[ \lim_{x \to 0} \frac{x}{x(\sqrt{x+2} + \sqrt{2})} \]

Cancel Common Factors

Since \(x\) is common in both the numerator and the denominator, it can be canceled, provided \(x eq 0\):\[ \lim_{x \to 0} \frac{1}{\sqrt{x+2} + \sqrt{2}} \]

Evaluate the Limit

Plug in \(x = 0\) into the expression:\[ \frac{1}{\sqrt{0+2} + \sqrt{2}} = \frac{1}{\sqrt{2} + \sqrt{2}} = \frac{1}{2\sqrt{2}} \]

Rationalize the Denominator (Optional)

For a fully simplified answer, rationalize the denominator by multiplying by \(\frac{\sqrt{2}}{\sqrt{2}}\):\[ \frac{\sqrt{2}}{4} \]

Key concepts

Rationalization

Rationalization is a process used in mathematics to eliminate a radical from the denominator or numerator of an expression. In the context of limit problems, like the one described above, rationalization is essential to simplify expressions so that they can be evaluated easily. Usually, one multiplies the given expression by a particular form of "1"—such as the conjugate of a binomial numerator or denominator—to simplify it.
Rationalizing the expression ensures that there are no radicals in the denominators or numerators, making calculations easier.
This step is vital for limits that initially seem indeterminate, as it allows you to more clearly see the limit behavior of the function. In our original problem, the numerator \(\sqrt{x+2} - \sqrt{2}\) was rationalized by multiplying by its conjugate, \(\sqrt{x+2} + \sqrt{2}\). This allowed us to simplify using the difference of squares.This method not only helps to simplify the expression but can also provide insights into the behavior of the function as \(x\) approaches the limit.

Difference of Squares

The difference of squares formula is a useful algebraic identity: \((a - b)(a + b) = a^2 - b^2\). This can be particularly useful when dealing with binomials involving radicals. In our exercise, the difference of squares was used to simplify the numerator:

• The expression \((\sqrt{x+2} - \sqrt{2})(\sqrt{x+2} + \sqrt{2})\) can be rewritten as \((x + 2) - 2\) using this identity.
• By doing this calculation, we transform a more complex expression into \(x\), which is much easier to work with, especially when evaluating limits.

Understanding the difference of squares helps simplify problems involving quadratic terms or expressions that fit this particular pattern. This knowledge ensures that you can identify and apply it quickly, which is a key part of accurately evaluating algebraic limits.

Conjugate Multiplication

Conjugate multiplication involves using a special form of the expression you're working with, specifically the "conjugate." A conjugate of \(a + b\) is \(a - b\), and vice-versa. It is primarily used to get rid of radicals in either the numerator or the denominator. In this exercise, we used the conjugate multiplication technique for the numerator \(\sqrt{x+2} - \sqrt{2}\) by multiplying by \(\sqrt{x+2} + \sqrt{2}\).

• This step transforms the numerator into a difference of squares, greatly simplifying the expression.
• It's an effective technique to neutralize complex radicals by creating a structure that can be simplified using basic algebraic identities.

This multiplication by its conjugate not only simplifies but also helps in later steps, particularly when dealing with indeterminate forms in limits. Always look for opportunities to apply conjugate multiplication, especially in problems involving radicals and limits.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that involves making an expression as easy as possible to work with, while maintaining its original value. For limits, simplification can often reveal the true behavior of functions as a variable approaches a specific value. In our example problem, after applying the difference of squares, the expression becomes:

• \(\lim_{x \to 0} \frac{x}{x(\sqrt{x+2} + \sqrt{2})}\)
• By canceling out the \(x\) in the numerator and denominator (considering \(x eq 0\)), you simplify the limit to \(\lim_{x \to 0} \frac{1}{\sqrt{x+2} + \sqrt{2}}\).

This simplification allows the limit to be evaluated directly, which in this case results in \(\frac{1}{2 \sqrt{2}}\). The process includes cleaning up the expression by removing common factors and unnecessary components, making it easier to understand and evaluate. Simplifying expressions is essential in calculus and beyond, ensuring you can efficiently solve and interpret mathematical problems.