Question

In Exercises 2.4.2-2.4.40, find the indicated limits. $$ \lim _{x \rightarrow \pi / 2}|\tan x|^{\cos x} $$

Short answer

Expression: \(\lim_{x \rightarrow \pi/2} |\tan x|^{\cos x}\) Answer: The limit as \(x\) approaches \(\frac{\pi}{2}\) of the given expression is \(1\).

Step-by-step solution

Analyze the base and exponent

As x approaches \(\pi/2\), the value of \(\tan x\) goes to infinity (\(\tan (\pi/2) = \infty\)). However, we're dealing with the absolute value, which means that \(|\tan x|\) also goes to infinity. As for \(\cos x\), it goes to zero as x approaches \(\pi/2\) since \(\cos (\pi/2) = 0\). So, we have an expression of the form \(a^b\) where \(a \rightarrow \infty\) and \(b \rightarrow 0\). To handle this situation, we will use logarithm technique.

Apply logarithm

Let's define a function $$ f(x) = |\tan x|^{\cos x} $$ Now take the natural logarithm (ln) of both sides: $$ \ln(f(x)) = (\cos x) \cdot \ln(|\tan x|) $$ The limit we are interested in will thus be transformed into $$ \lim_{x \rightarrow \pi/2} \ln(f(x)) $$

Use L'Hôpital's Rule

Now, rewrite our limit as $$ \lim_{x \rightarrow \pi/2} \frac{\ln(|\tan x|)}{1/\cos x} $$ The limit has the form \((\infty) / (\infty)\), so we can use L'Hôpital's Rule. We will take the derivative of the numerator and the denominator. The derivative of the numerator, $$ \frac{d}{dx} \ln(|\tan x|) = \frac{\sec^2 x}{\tan x} $$ The derivative of the denominator, $$ \frac{d}{dx} \frac{1}{\cos x} = \sin x $$ Now, apply L'Hôpital's rule: $$ \lim_{x \rightarrow \pi/2} \frac{\sec^2 x}{\tan x \cdot \sin x} $$

Analyze the new limit

We can rewrite \(\sec^2 x = \frac{1}{\cos^2 x}\) which implies $$ \lim_{x \rightarrow \pi/2} \frac{1}{\cos^2 x \cdot \sin x \cdot \tan x} $$ As \(x\) approaches \(\pi/2\), \(\cos x\) goes to 0, \(\tan x\) goes to infinity and \(\sin x\) goes to 1. So the limit becomes: $$ \lim_{x \rightarrow \pi/2} \frac{1}{0 \cdot 1 \cdot \infty} = 0 $$ Since the limit of \(\ln(f(x))\) as \(x\) approaches \(\pi/2\) is 0, then the limit of the original function will be: $$ \lim_{x \rightarrow \pi/2} |\tan x|^{\cos x} = e^0 = \boxed{1}. $$

Key concepts

L'Hôpital's Rule

When solving limits that result in indeterminate forms like \(0/0\) or \(\infty/\infty\), L'Hôpital's Rule is an essential tool. It states that if the limits of functions \( f(x) \) and \( g(x) \) both approach 0 or \(\infty\), and they are differentiable in a neighborhood of a point \(c\) (except possibly at \(c\) itself), then \( \lim_{x \to c} (f(x)/g(x)) \) can be found by computing \( \lim_{x \to c} (f'(x)/g'(x)) \), assuming this limit exists or equals \( \infty \).

When we apply this rule, we differentiate the numerator and denominator separately. The power of L'Hôpital's Rule lies in its ability to simplify complicated limits into more solvable forms. However, it should only be used when necessary conditions are met and if no simpler methods are available. By transforming the limit into a derivative, we often encounter a simpler expression that approaches a real number or \( \infty \).

Natural Logarithm

The natural logarithm, denoted as \(\ln(x)\), plays a crucial role in finding limits, especially when dealing with exponential functions. It is the logarithm to the base \(e\), where \(e\) is an irrational and transcendental constant approximately equal to 2.71828.

The natural logarithm helps to simplify multiplication into addition, powers into products, and is the inverse of the natural exponential function. To apply it to limits, we took the logarithm of the function and then found the limit. After evaluating the limit of the logarithmic function, we can transform it back using the exponential function. This approach is particularly helpful when dealing with limits involving forms like \(0 \cdot \infty\) or \(\infty^0\) since it can transform the limit into one that evaluates more directly.

Trigonometric Limits

When working with trigonometric limits, we often encounter behavior that involves angles approaching critical points where their sine, cosine, and tangent values take on infinite or zero limits. Understanding the fundamental trigonometric limits, such as \(\lim_{x \to 0} (\sin(x)/x) = 1\) and \(\lim_{x \to 0} (\tan(x)/x) = 1\), is key to evaluating more complex expressions.

In the context of our problem, we dealt with \(\tan x\) and \(\cos x\) as \(x\) approached \(\pi/2\). These relations can give rise to indeterminate forms when taken to limits involving infinity or zero. Knowing how these functions behave as they approach these critical angles and how they relate to other trigonometric functions through identities enriches our understanding and ability to evaluate difficult limits.