Problem 15 - Calculus: Early Transcendentals

Question

Compute the following limits. $$ \lim _{x \rightarrow \pi / 2} \frac{\cos x}{(\pi / 2)-x} $$

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Identify the Limit Form

We have the limit \( \lim_{x \rightarrow \pi / 2} \frac{\cos x}{(\pi / 2) - x} \). As \( x \rightarrow \pi/2 \), \( \cos x \rightarrow 0 \) and \( (\pi/2) - x \rightarrow 0 \). This is a \( \frac{0}{0} \) indeterminate form, allowing us to use L'Hôpital's Rule.

Step 2: Apply L'Hôpital's Rule

According to L'Hôpital's Rule, if we have an indeterminate form \( \frac{0}{0} \), we can differentiate the numerator and the denominator separately. Differentiate \( \cos x \) to get \( -\sin x \). Differentiate \( (\pi/2) - x \) to get \( -1 \).

Step 3: Compute the Differentiated Limit

The limit becomes: \[ \lim_{x \rightarrow \pi/2} \frac{-\sin x}{-1} = \lim_{x \rightarrow \pi/2} \sin x. \]

Step 4: Evaluate the Limit

Now calculate \( \lim _{x \rightarrow \pi / 2} \sin x \). Since \( \sin(\pi/2) = 1 \), the limit evaluates to 1.

Key Concepts

Limits in Calculus

Understanding limits in calculus is essential for grasping the behavior of functions as they approach specific points. When we talk about limits, we're interested in what value a function approaches as the input gets closer to a certain point. They help us analyze:

• The behavior of functions at points where they may not be well-defined.
• How functions behave as they tend toward infinity.

The notation \( \lim_{x \to a} f(x) \) means "the limit of \( f(x) \) as \( x \) approaches \( a \)." This concept is vital in calculus because it lays the groundwork for derivatives and integrals. In our example, we calculated \( \lim _{x \rightarrow \pi / 2} \frac{\cos x}{(\pi / 2) - x} \). This limit tells us how the function behaves as \( x \) becomes very close to \( \pi / 2 \). Limits are crucial for determining continuity and understanding infinite processes.

Indeterminate Forms

Indeterminate forms appear when a straightforward evaluation of a limit results in expressions like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). These forms do not directly lead to a clear answer. Instead, they require special techniques to evaluate properly. In our problem, as \( x \to \pi/2 \), both the numerator \( \cos x \) and the denominator \( (\pi/2) - x \) approach zero.
This results in a \( \frac{0}{0} \) indeterminate form.
To resolve such forms, L'Hôpital's Rule is particularly helpful. This rule states that if \( \lim_{x \to a} \frac{f(x)}{g(x)} \) is indeterminate, we can compute \( \lim_{x \to a} \frac{f'(x)}{g'(x)} \), provided that this new limit exists.

• Differentiating changes the form of the limit but retains the essential behavior of the original functions.
• Helps in determining the actual behavior of complex expressions.

Trigonometric Limits

Trigonometric limits specifically involve functions like sine, cosine, and tangent as they approach particular angles. Many limits in calculus involve trigonometric functions— understanding their behavior is crucial.
Sine and cosine, for instance, have well-known values at critical angles like 0, \( \frac{\pi}{2} \), and \( \pi \).
In our example, the limit involves transforming an indeterminate form into a trigonometric expression.

• By applying L'Hôpital's Rule, \( \lim_{x \to \pi/2} \frac{\cos x}{(\pi / 2) - x} \) simplified to \( \lim_{x \to \pi/2} \sin x \).
• As \( x \to \pi/2 \), it is important to know that \( \sin x \) approaches 1, which helped us evaluate the limit correctly.

Trigonometric identities and properties often simplify the process of solving limits, making them easier to analyze and compute.