Problem 370 - Calculus Volume 1

Question

For the following exercises, evaluate the limits with either L'Hôpital's rule or previously learned methods. $$ \lim _{x \rightarrow \pi / 2} \frac{\cos x}{2} $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Identify the Expression

We need to evaluate or simplify: \(\lim _{x \rightarrow \pi / 2} \frac{\cos x}{2}\).

Step 2: Apply Order of Operations

Following the order of operations (PEMDAS/BODMAS), we simplify the expression step by step, handling parentheses, exponents, multiplication/division, then addition/subtraction.

Step 3: Compute the Result

Carrying out the operations systematically, we arrive at the result.

Step 4: State the Result

The limit is 0.

Key Concepts

L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool in calculus used to evaluate limits which result in indeterminate forms. It's especially helpful for forms like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), where trying to directly substitute the limiting value into the function doesn't work. Instead of being stuck, we can use L'Hôpital's Rule, which states that: If \[ \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{0}{0} \text{ or } \frac{\infty}{\infty}, \] then \[ \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \] provided that this latter limit exists.This means we take the derivative of the numerator, \(f(x)\), and the derivative of the denominator, \(g(x)\), and evaluate the limit of their quotient. This rule simplifies the process significantly in cases where conventional methods do not apply. Keep in mind that L'Hôpital's Rule can be applied iteratively until the limit is resolved.

Direct Substitution

Direct substitution is one of the simplest methods for finding limits. It involves simply substituting the value that \(x\) approaches directly into the function. If the function is continuous and well-defined at that point, this basic method provides a quick and efficient way to evaluate limits. For example, in the original problem, we wanted to find \[ \lim _{x \rightarrow \pi / 2} \frac{\cos x}{2}. \]By substituting \(x = \pi/2\) into the function, we evaluate \[ \frac{\cos(\pi/2)}{2} = \frac{0}{2} = 0. \]Here, direct substitution worked like a charm because cosine of \(\pi/2\) equals zero, making the process incredibly straightforward. However, this technique is most effective when the result is not an indeterminate form or when the function remains defined as \(x\) approaches the value.

Indeterminate Forms

Indeterminate forms arise in calculus when evaluating a limit doesn't initially provide a direct answer. Commonly, indeterminate forms are seen as expressions like \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), \(0 \cdot \infty\), \(\infty - \infty\), \(1^{\infty}\), \(0^0\), and \(\infty^0\). These forms essentially tell us that more work is needed and that the limit cannot be resolved by direct substitution alone.In these cases, we need to apply additional methods, with L'Hôpital's Rule being a notable option, especially for \(\frac{0}{0}\) and \(\frac{\infty}{\infty}\). When dealing with indeterminate forms:

• Determine whether the function can be simplified by factoring or rationalizing.
• Consider using L'Hôpital's Rule to take derivatives for a clearer limit determination.
• Sometimes, algebraic manipulation or trigonometric identities can also resolve these forms.

Understanding indeterminate forms is crucial because they highlight the limits requiring extra attention to uncover the definitive answer rather than arbitrary outcomes.