Question

\(\operatorname{Lt}_{x \rightarrow b} \frac{\sqrt{x}-\sqrt{b}}{x-b}=\) (1) \(\frac{1}{2 \mathrm{~b}}\) (2) \(\frac{\sqrt{2}}{\mathrm{~b}}\) (3) \(\frac{1}{2 \sqrt{\mathrm{b}}}\) (4) None of these

Short answer

Answer: (3) \(\frac{1}{2 \sqrt{\mathrm{b}}}\)

Step-by-step solution

Identify the given expression

We are given the expression to find the limit: \(\operatorname{Lt}_{x \rightarrow b} \frac{\sqrt{x}-\sqrt{b}}{x-b}\)

Rationalize the numerator

To rationalize the numerator, multiply both the numerator and the denominator by the conjugate of the numerator: \(\frac{\sqrt{x}+\sqrt{b}}{\sqrt{x}+\sqrt{b}}\) The modified expression would be: \(\operatorname{Lt}_{x \rightarrow b} \frac{(\sqrt{x}-\sqrt{b})(\sqrt{x}+\sqrt{b})}{(x-b)(\sqrt{x}+\sqrt{b})}\)

Simplify the numerator and denominator

Using the difference of squares, we can simplify the numerator: \((\sqrt{x})^2-(\sqrt{b})^2 = x - b\) The new modified expression would be: \(\operatorname{Lt}_{x \rightarrow b} \frac{x - b}{(x-b)(\sqrt{x}+\sqrt{b})}\)

Cancel out common factors

We can cancel out \((x - b)\) from both the numerator and the denominator, which gives us: \(\operatorname{Lt}_{x \rightarrow b} \frac{1}{\sqrt{x}+\sqrt{b}}\)

Evaluate the limit as x approaches b

Now, evaluate the limit as x approaches b: \(\operatorname{Lt}_{x \rightarrow b} \frac{1}{\sqrt{x}+\sqrt{b}} = \frac{1}{\sqrt{b}+\sqrt{b}} = \frac{1}{2\sqrt{b}}\) The limit of the given expression as x approaches b is \(\frac{1}{2\sqrt{b}}\). Therefore, the correct answer is (3) \(\frac{1}{2 \sqrt{\mathrm{b}}}\).

Key concepts

Rationalizing the Numerator

Rationalizing the numerator is an algebraic technique used to eliminate radicals from the top of a fraction. This is often done by multiplying the numerator by its conjugate. The conjugate of \( \sqrt{x} - \sqrt{b} \) is \( \sqrt{x} + \sqrt{b} \), which creates a difference of squares when multiplied by the original numerator.

Why do we rationalize? Rationalizing the numerator can help in simplifying an expression, especially when evaluating limits where direct substitution leads to an indeterminate form like \( \frac{0}{0} \). By doing this, we get rid of the radicals which can make further simplification or evaluation much easier. In our exercise, rationalizing makes the limit solvable by canceling out common factors.

Difference of Squares

The difference of squares is an algebraic pattern that emerges when you subtract two squares, i.e., \( a^2 - b^2 \). This expression can be factored into \( (a+b)(a-b) \). This property is vital for simplifying algebraic expressions and is particularly handy in calculus when dealing with limits.

When rationalizing the numerator, we often use the difference of squares to simplify the expression. In the provided exercise, the pattern \( (\sqrt{x})^2 - (\sqrt{b})^2 \) helped us to rewrite the numerator from a radical form into a linear expression, which allowed for the subsequent algebraic simplifications.

Evaluating Limits

Evaluating limits is a fundamental concept in calculus, where we determine the value that a function approaches as the input approaches a particular point. Limits can be evaluated through direct substitution, factoring, rationalization, using theorems, or special rules like L'Hôpital's rule when applicable.

In our exercise, after rationalization and simplifying using the difference of squares, evaluating the limit of \( \frac{1}{\sqrt{x} + \sqrt{b}} \) as \( x \) approaches \( b \) is straightforward. The function is continuous at \( x=b \) and direct substitution gives us the result.

Algebraic Simplification

Algebraic simplification refers to the process of reducing expressions to their simplest form. This involves factoring, expanding, combining like terms, and canceling out common factors. Simplification can often make evaluation of expressions clearer and is particularly useful in solving limits, as it can reveal the behavior of the function as it approaches a specific point.

In our step-by-step solution, algebraic simplification involved canceling out the common \( (x - b) \) term, which resulted in a much simpler expression that could easily be evaluated as \( x \) approaches \( b \) to find the limit.