Question

Find the limit as \(x \rightarrow 0\) of \(\left[\sqrt{1+x^{m}}-\sqrt{1-x^{m}}\right] / x^{n}\), in which \(m\) and \(n\) are positive integers.

Short answer

0 if \(m > n\), 1 if \(m = n\), infinite if \(m < n\).

Step-by-step solution

Expression Setup

Given the expression \(\frac{\text{ }\text{ }\text{ }\bigl(\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\bigl(\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ ... \)\big]\big/ x^{n} \ where \(x\rightarrow0\).

Apply Binomial Expansion

Since \(x\rightarrow0\) and \(m\) is a positive integer, use binomial expansion for small values of \(x\). Expand \(\sqrt{1+x^{m}}\) and \(\sqrt{1-x^{m}}\): \[ \sqrt{1+x^{m}} \approx 1 + \frac{x^{m}}{2} \text{ and } \sqrt{1-x^{m}} \approx 1 - \frac{x^{m}}{2}. \]

Substitute the Expansions

Substitute the binomial expansions into the original expression: \[ \frac{\bigl(1 + \frac{x^{m}}{2} - (1 - \frac{x^{m}}{2})\bigr)}{x^{n}} = \frac{x^{m}}{x^{n}}. \]

Simplify the Fraction

Simplify the fraction: \[ \frac{x^{m}}{x^{n}} = x^{m-n}. \]

Evaluate the Limit

Evaluate the limit as \(x \rightarrow 0\): \[ \lim_{x \rightarrow 0} x^{m-n}. \] This limit depends on the relationship between \(m\) and \(n\):- If \(m = n\), the limit is 1.- If \(m > n\), the limit is 0.- If \(m < n\), the limit is infinite.

Key concepts

Binomial Expansion

Binomial expansion is a technique used to expand expressions raised to a power. When dealing with binomial expansion for small values of a variable, such as in our problem where \(x \rightarrow 0\), a Taylor series approximation is used. This helps in simplifying the expressions so we can evaluate the limit more easily.

For example, if we take \(\sqrt{1+x^m}\) and use binomial expansion, we get:

\( \sqrt{1 + x^m} \approx 1 + \frac{x^m}{2} \) and similarly for \(\sqrt{1 - x^m} \approx 1 - \frac{x^m}{2}\).
These approximations work well for small values of \(x\), and they make it easier to solve limits involving square roots and other such expressions.

Positive Integers

In the given problem, \(m\) and \(n\) are positive integers, which simplifies our calculations.

Positive integers are whole numbers greater than zero (e.g., 1, 2, 3, ...).

Understanding that \(m\) and \(n\) are positive is crucial because it ensures that our expansions and simplifications hold true. For instance, it confirms that binomial expansions we used are valid within the range of small \(x\)-values. This positiveness guarantees the non-negative behavior of the expressions \(x^m\) and \(x^n\).

So, throughout the problem, we leverage their positive values to finally determine the limit depending on the relationship between these two integers.

Limit Evaluation

Limit evaluation is one of the core concepts in calculus. It helps us understand the behavior of a function as the variable approaches a certain value.

In this problem, we want to find the limit of an expression as \(x\) approaches 0. We've simplified our expression to \(x^{m-n}\), so we can now look at three different cases based on the relationship between \(m\) and \(n\):

• If \(m = n\): The expression \(x^{m-n}\) becomes \(x^0\), which equals 1. Thus, the limit is 1.
• If \(m > n\): The exponent \(m-n\) is positive, so as \(x\) approaches 0, \(x^{m-n}\) approaches 0. The limit is 0.
• If \(m < n\): The exponent \(m-n\) is negative, which means \(x^{m-n}\) will grow very large as \(x\) approaches 0, making the limit infinity.

This evaluation allows us to confidently conclude the behavior of the function based on the values of \(m\) and \(n\).

Understanding and knowing how to evaluate limits involves recognizing these scenarios and applying the appropriate solution based on the algebraic manipulation of the given function.