Problem 69 - Thomas Calculus

Question

L'Hopital's Rule does not help with the limits in Exercises \(67-74 .\) Try it- you just keep on cycling. Find the limits some other way. $$ \lim _{x \rightarrow(\pi / 2)^{-}} \frac{\sec x}{\tan x} $$

Step-by-Step Solution

Answer

The limit is 1 as \( x \to (\pi/2)^{-}.\)

Step 1: Understand the given limit expression

The limit expression provided is \( \lim_{x \to (\pi/2)^{-}} \frac{\sec x}{\tan x} \). This means we are looking for the limit of the function as \( x \) approaches \( \pi/2 \) from the left side (i.e., values slightly less than \( \pi/2 \)).

Step 2: Rewrite the trigonometric terms using their identities

Remember that \( \sec x = \frac{1}{\cos x} \) and \( \tan x = \frac{\sin x}{\cos x} \). Substituting these into the limit expression gives:\[ \lim_{x \to (\pi/2)^{-}} \frac{\frac{1}{\cos x}}{\frac{\sin x}{\cos x}} = \lim_{x \to (\pi/2)^{-}} \frac{1}{\sin x} \].

Step 3: Analyze the behavior of the sine function near \( \pi/2 \)

As \( x \to (\pi/2)^{-} \), \( \sin x \to 1 \). Thus, as \( x \) approaches \( \pi/2 \) from the left, \( \sin x \) gets closer and closer to 1.

Step 4: Evaluate the simplified limit expression

Given that \( \sin x \to 1 \) as \( x \to (\pi/2)^{-} \), the expression \( \frac{1}{\sin x} \to \frac{1}{1} = 1 \). Therefore, the limit is \( 1 \).

Key Concepts

L'Hopital's Rule

L'Hopital's Rule is a powerful tool in calculus used for finding limits of indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). The rule states that for functions \( f(x) \) and \( g(x) \) where \( \lim_{x \to c} f(x) = 0 \) and \( \lim_{x \to c} g(x) = 0 \) or both functions approach infinity as \( x \to c \), the following holds:\[\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}\]if the right-hand limit exists.However, this rule does not always work well for every situation, as seen in the original problem. There are cases where applying L'Hopital's more than once leads to cycling back to the original limit form, or the derivatives might not exist or be complex. In such cases, alternative strategies like trigonometric identities or direct substitution become crucial.

Trigonometric Identities

Trigonometric identities are mathematical equations that relate various trigonometric functions. These identities are tremendously useful for simplifying complex trigonometric expressions or finding limits.For the given exercise, understanding key identities is necessary:

• \( \sec x = \frac{1}{\cos x} \)
• \( \tan x = \frac{\sin x}{\cos x} \)

By substituting these identities into the original limit expression, it simplifies the task of finding limits. The expression \( \frac{\sec x}{\tan x} \) transforms into \( \frac{1}{\sin x} \), which is far easier to analyze as \( x \) approaches \( \pi/2 \). Applying these identities helps to avoid pitfalls and directly leads us to evaluate more straightforward parts of the function near the limit point.

Sine Function

The sine function, denoted as \( \sin x \), is a periodic function with a range between -1 and 1. As \( x \) approaches \( \pi/2 \), \( \sin x \) heads closer to its maximum value of 1.In our exercise's context, we evaluate \( \lim_{x \to (\pi/2)^{-}} \frac{1}{\sin x} \). As \( x \) nears \( \pi/2 \) from the left, imagine that values are slightly less than \( \pi/2 \) (like \( \pi/2 - 0.1 \)), and you compute \( \sin(\pi/2 - 0.1) \), which will be very close to 1. Therefore, the concept here is about recognizing behavior patterns of the sine function as it reaches the limit, allowing you to deduce the behavior of the whole expression.

Approaching from Left

"Approaching from the left" indicates that we are evaluating the behavior of a function when the variable \( x \) is incrementally getting close to a particular point from lesser values. It is expressed mathematically as \( x \to c^{-} \). In the original exercise, \( x \to (\pi/2)^{-} \) means inspecting the limit as \( x \) comes from values just smaller than \( \pi/2 \). This is essential in ensuring how the function behaves precisely at its boundary; as \( x \) is not reaching \( \pi/2 \) from the right side or exactly at \( \pi/2 \).Understanding the concept of approaching from one side helps ensure your analysis accurately reflects real-world conditions where functions could behave differently when coming from different sides or directions. This differentiation allows you to achieve complete and thorough solutions while solving limits.