Question

\(\lim _{\mathrm{x} \rightarrow 0}\left[\left\\{\sqrt{\left.\left. \left.\left(5+\mathrm{x}^{5}\right)-\sqrt{(5-\mathrm{x}}^{5}\right)\right\\} / \mathrm{x}^{5}\right]}=?\right.\right.\) (a) 5 (b) 25 (c) \((\sqrt{5})^{-1}\) (d) \(4 \sqrt{5}\)

Short answer

(c) \((\sqrt{5})^{-1}\)

Step-by-step solution

Rewrite and simplify the expression using difference of squares formula

The expression inside the limit is \(\frac{\sqrt{(5+x^5)} - \sqrt{(5-x^5)}}{x^5}\). We can simplify this expression by multiplying it by its conjugate, which is \(\frac{\sqrt{(5+x^5)} + \sqrt{(5-x^5)}}{\sqrt{(5+x^5)} + \sqrt{(5-x^5)}}\). The new expression becomes: \(\frac{(\sqrt{(5+x^5)} - \sqrt{(5-x^5)})(\sqrt{(5+x^5)} + \sqrt{(5-x^5)})}{x^5(\sqrt{(5+x^5)} + \sqrt{(5-x^5)})}\)

Simplify the numerator using the difference of squares formula

Applying the difference of squares formula, the numerator simplifies to: \((5+x^5)-(5-x^5)\) Which further simplifies to: \(2x^5\)

Simplify the entire expression

Now, substitute the simplified numerator back into the expression: \(\frac{2x^5}{x^5(\sqrt{(5+x^5)}+\sqrt{(5-x^5)})}\) We can cancel out the \(x^5\) terms: \(\frac{2}{\sqrt{(5+x^5)}+\sqrt{(5-x^5)}}\)

Evaluate the limit

Now, evaluate the limit as x approaches 0: \(\lim_{x\to 0}\frac{2}{\sqrt{(5+0^5)}+\sqrt{(5-0^5)}}\) Which simplifies to: \(\frac{2}{\sqrt{5}+\sqrt{5}} = \frac{2}{2\sqrt{5}}\) Finally, we get: \(\frac{1}{\sqrt{5}}\) The answer is (c) \((\sqrt{5})^{-1}\).

Key concepts

Difference of Squares

One of the key techniques used in solving calculus limit problems is the difference of squares. This concept is very handy when dealing with expressions involving square roots. The difference of squares formula is an algebraic identity that states: \(a^2 - b^2 = (a-b)(a+b)\). This allows us to factor complex expressions into simpler components. For the given problem, the expression inside the limit involves square roots, \(\frac{\sqrt{(5+x^5)}-\sqrt{(5-x^5)}}{x^5}\). By introducing the technique of multiplication by the conjugate, we multiply the numerator and the denominator by \(\sqrt{(5+x^5)}+\sqrt{(5-x^5)}\). This transforms the numerator into a difference of squares: \( (\sqrt{(5+x^5)})^2 - (\sqrt{(5-x^5)})^2 = (5+x^5) - (5-x^5)\), which simplifies straightforwardly to \(2x^5\). Thus, using the difference of squares is an effective way to simplify the expression and make it manageable.

Simplifying Expressions

Simplifying expressions is crucial in calculus, particularly when dealing with limits. It often involves factoring, expanding, or other algebraic manipulations to make the expression easier to work with. In this exercise, after applying the difference of squares formula, the numerator of the expression becomes \(2x^5\).To progress further:

• Substitute the simplified numerator back into the original fraction, giving \(\frac{2x^5}{x^5(\sqrt{(5+x^5)}+\sqrt{(5-x^5)})}\).
• Notice that the \(x^5\) terms in both the numerator and the denominator can be cancelled out, leaving \(\frac{2}{\sqrt{(5+x^5)}+\sqrt{(5-x^5)}}\).

The cancellation significantly simplifies the expression, which is now much easier to evaluate. Simplifying expressions is an essential step in solving calculus problems since it often reveals the underlying straightforward form of what initially appears as a complex equation.

Evaluating Limits

Evaluating limits is a fundamental concept in calculus, where the focus is on understanding the behavior of functions as they approach a particular point. The concept of a limit is essential for defining derivatives and integrals, which are core components of calculus. For this exercise, the limit is evaluated as \(x\) approaches zero in the expression \(\frac{2}{\sqrt{(5+x^5)}+\sqrt{(5-x^5)}}\).Here's how to evaluate it:

• Observe how the terms within the square roots tend towards constant values when \(x=0\). Specifically, \(\sqrt{5+x^5}\) and \(\sqrt{5-x^5}\) both approach \(\sqrt{5}\).
• The denominator simplifies to \(\sqrt{5} + \sqrt{5} = 2\sqrt{5}\).
• The limit then becomes \(\lim_{x \to 0} \frac{2}{2\sqrt{5}}=\frac{1}{\sqrt{5}}\).

By following these steps, you determine that the correct value of the limit is \(\frac{1}{\sqrt{5}}\), which matches the multiple-choice answer \(c) (\sqrt{5})^{-1}\). Evaluating limits accurately forms the foundation for understanding how functions behave at specific points, providing insight into their continuity and differentiability.