Question

Evaluate the following limits. $$\lim _{x \rightarrow 0^{+}}(\sin x) \sqrt{\frac{1-x}{x}}$$

Short answer

Answer: The limit of the expression is \(\infty\).

Step-by-step solution

Rewrite the expression in a more manageable form

In this step, we're going to rewrite the expression as follows: $$\lim_{x\to 0^+}\frac{\sin{x}}{\sqrt{\frac{x}{1-x}}}$$ By performing this transformation, we can focus on simplifying and evaluating the limit of the expression in a more straightforward manner.

Apply L'Hôpital's Rule to the expression

In order to apply L'Hôpital's Rule and solve the limit, we need to first rewrite the expression in terms of a quotient of functions. This can be accomplished by performing the following steps: $$\lim_{x\to 0^+}\frac{\sin{x}}{\sqrt{\frac{x}{1-x}}} = \lim_{x\to 0^+}\frac{\sin{x}(1-x)}{\sqrt{x}}$$ Now, we need to differentiate both the numerator and the denominator with respect to x: $$\lim_{x\to 0^+}\frac{(\cos{x})(1-x)-\sin{x}}{\frac{1}{2\sqrt{x}}}$$

Evaluate the limit

Now, we can plug in x = 0 to the expression: $$\lim_{x\to 0^+}\frac{(\cos{x})(1-x)-\sin{x}}{\frac{1}{2\sqrt{x}}} = \frac{(\cos{0})(1-0)-\sin{0}}{\frac{1}{2\sqrt{0}}} = \frac{1}{0}$$ Since we have a limit that approaches infinity as x approaches 0 from the positive side, the limit of the expression is: $$\lim _{x \rightarrow 0^{+}}(\sin x) \sqrt{\frac{1-x}{x}} = \infty$$

Key concepts

L'Hôpital's Rule

L'Hôpital's Rule is a key technique in calculus for finding the limits of indeterminate forms such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). It states that if the limits of both the numerator and the denominator of a function are 0 or both are infinite as x approaches a particular value, then the limit of the function can be found by taking the derivatives of the numerator and denominator separately.

Applied correctly, it can turn a challenging limit evaluation into a simpler problem. For our textbook exercise, L'Hôpital's Rule was invoked to evaluate the limit of the given expression as x approaches 0 from the positive side. By differentiating the numerator and denominator, we simplified the expression, allowing us to find that the limit is infinity. Care must be taken to continuously check if the form is still indeterminate after taking the derivatives; if it's not, then the limit can be evaluated using the new expression.

Indeterminate Forms

Indeterminate forms arise when evaluating limits and include expressions like \( 0/0 \), \( \infty/\infty \), \( 0 \cdot \infty \), \( \infty - \infty \), \( 0^0 \), \( \infty^0 \), and \( 1^\infty \). These forms don't lead to conclusive results without further analysis or manipulation. In our exercise, when initially attempting to evaluate the limit, we'd reach an expression that simplifies to \( 1/0 \), which is another type of infinity-related indeterminate scenario.

In the process of applying L'Hôpital's Rule to our exercise, we ensure the expression is correctly identified as an indeterminate form to which the rule can be applied. This identification is crucial as applying the rule to an expression that is not an indeterminate form can result in incorrect conclusions.

Limit Evaluation

Limit evaluation is a fundamental concept in calculus, which involves finding the value that a function approaches as the input approaches a specified value. Limits can be evaluated from the right (\(0^+\)) or from the left (\(0^-\)), and it's pertinent to differentiate between the two, as they can yield different results.

This calculation is the groundwork for defining derivatives and integrals. In our exercise, after applying L'Hôpital's Rule, we found the limit approach infinity as x approached 0 from the right. The approach was systematic: apply necessary calculus rules, simplify, and then plug in the limit value. It’s essential for students not just to memorize the steps, but to understand why each step is taken to ensure they can apply similar reasoning to different problems.

Simplifying Expressions

Simplifying expressions is the process of altering a mathematical expression to make it easier to understand or work with, without changing its value. This can include factoring, combining like terms, and reducing fractions. In the context of limit evaluation, simplification might also involve changing the form of the function to apply specific rules like L'Hôpital’s.

During the evaluation of our exercise limit, simplification was a pivotal step. The initial expression was transformed into a ratio, facilitating the use of L'Hôpital’s Rule. This highlights the usefulness of simplifying expressions as a strategy for making an otherwise complex limit manageable. It is an iterative and creative process, where several reformulations might be necessary to reach a solvable form. Patience and practice are essential to master this skill in calculus.