Problem 3 - Essential Calculus Early Transcendentals

Question

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.$$\lim _{x \rightarrow(\pi / 2)^{+}} \frac{\cos x}{1-\sin x}$$

Step-by-Step Solution

Answer

The limit is \(+\infty\).

Step 1: Determine the Indeterminate Form

To evaluate the limit \(\lim _{x \rightarrow (\pi / 2)^{+}} \frac{\cos x}{1-\sin x}\), first substitute \(x = \frac{\pi}{2}\). We have \(\cos\left(\frac{\pi}{2}\right) = 0\) and \(1 - \sin\left(\frac{\pi}{2}\right) = 0\), which results in the undefined \(\frac{0}{0}\) form.

Step 2: Apply L'Hospital's Rule

Since we have a \(\frac{0}{0}\) indeterminate form, we can apply L'Hospital's Rule, which allows us to take the derivative of the numerator and the denominator separately. The derivative of \(\cos x\) is \(-\sin x\), and the derivative of \(1 - \sin x\) is \(-\cos x\). Therefore, the limit becomes \(\lim _{x \rightarrow (\pi / 2)^{+}} \frac{-\sin x}{-\cos x}\).

Step 3: Simplify the Limit after Differentiation

Simplify the limit expression \(\frac{-\sin x}{-\cos x}\) to \(\frac{\sin x}{\cos x}\), which is the tangent of \(x\). Thus, the limit transforms into \(\lim _{x \rightarrow (\pi / 2)^{+}} \tan x\).

Step 4: Evaluate the New Limit Form

As \(x\) approaches \(\frac{\pi}{2}\) from the right, \(\tan x\) approaches positive infinity because the tangent function has a vertical asymptote at \(\frac{\pi}{2}\) from the right side. Therefore, the limit is \(+\infty\).

Key Concepts

Indeterminate Forms

In calculus, while evaluating limits, we often encounter forms such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), known as indeterminate forms. These forms arise when direct substitution of the limit value into the function leads to an undefined situation.
They don't provide enough information alone to determine the limit's value. In these cases, additional techniques like simplification or rules such as l'Hospital's Rule are needed to resolve the limit.

For example, in our given exercise with \( \lim _{x \rightarrow (\pi / 2)^{+}} \frac{\cos x}{1-\sin x} \), direct substitution results in \( \frac{0}{0} \). This is a classic example of an indeterminate form, prompting us to apply further techniques to find the limit's true behavior.

Calculus Limits

Calculus limits help us understand the behavior of functions as they approach certain points or tend toward infinity. This concept is foundational for calculus, as it is used to define derivatives and integrals.
A limit is essentially the value that a function approaches as the input approaches some point. In formal terms, the limit of a function \( f(x) \) as \( x \) approaches \( c \) is \( L \), written as \( \lim_{x \to c} f(x) = L \), if \( f(x) \) becomes arbitrarily close to \( L \) for \( x \) sufficiently close to \( c \).

For the exercise given, we are interested in finding the limit as \( x \) approaches \( \pi/2 \) from the right. This kind of right-hand limit looks at the behavior of \( f(x) \) as \( x \) approaches a point from higher values, providing valuable insights into the nature of functions that are not continuous or smooth at that point.

Trigonometric Limits

Trigonometric limits often involve functions like sine, cosine, and tangent. These functions exhibit unique properties, especially as they approach certain well-known angles, making them particularly interesting in limit problems.
For instance, the function \( \tan x \) becomes undefined at odd multiples of \( \frac{\pi}{2} \) due to vertical asymptotes, indicating tangent's tendency toward infinity at these points when approached from the right or left.

In our exercise, once we apply l'Hospital's Rule to resolve the \( \frac{0}{0} \) indeterminate form, we simplify it to \( \frac{\sin x}{\cos x} \), which is \( \tan x \). Evaluating the right-hand limit of \( \tan x \) as \( x \) approaches \( \frac{\pi}{2} \), the function's value tends toward positive infinity, demonstrating the utility of understanding trigonometric functions' behaviors in limit calculations.