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Chapter 4: Problem 88

Use analytical methods to evaluate the following limits. $$\lim _{x \rightarrow \pi / 2}(\pi-2 x) \tan x$$

Short Answer

Expert verified
Question: Evaluate the limit of the function (\(\pi-2x\) )(\(\tan x\)) as x approaches \(\frac{\pi}{2}\). Answer: The limit of the given expression as \(x\) approaches \(\frac{\pi}{2}\) is 0: $$ \lim_{x \rightarrow \frac{\pi}{2}}(\pi-2 x) \tan x = 0 $$

Step by step solution

01

Identify the Functions

The given expression consists of two functions: 1. \(f(x) = \pi - 2x\) 2. \(g(x) = \tan x\) We will first find the limits of these two functions as \(x\) approaches \(\frac{\pi}{2}\).
02

Find the limit of f(x)

To find the limit of \(f(x)\) as \(x\) approaches \(\frac{\pi}{2}\), simply replace \(x\) with \(\frac{\pi}{2}\): $$ \lim_{x \rightarrow \frac{\pi}{2}}(\pi - 2x) = \pi - 2\left(\frac{\pi}{2}\right) = 0 $$
03

Find the limit of g(x)

To find the limit of \(g(x)\) as \(x\) approaches \(\frac{\pi}{2}\), notice that \(\tan x\) tends to infinity as \(x\) approaches \(\frac{\pi}{2}\). Therefore, the limit of \(g(x)\) is: $$ \lim_{x \rightarrow \frac{\pi}{2}}(\tan x) = \infty $$
04

Use the properties of limits to evaluate the given limit

Now, we have the limits of both functions as \(x\) approaches \(\frac{\pi}{2}\). We will multiply these limits to find the given limit: $$ \lim_{x \rightarrow \frac{\pi}{2}}(\pi-2 x) \tan x = \lim_{x \rightarrow \frac{\pi}{2}}(\pi-2 x) \cdot \lim_{x \rightarrow \frac{\pi}{2}} \tan x $$ Substituting the limits we found earlier: $$ = 0 \cdot \infty $$ The product \(0 \cdot \infty\) is an indeterminate form, which cannot be directly computed.
05

Apply L'H么pital's Rule

To handle the indeterminate form, we will apply L'H么pital's Rule. To do that, we will differentiate both the numerator and the denominator with respect to \(x\): $$ \frac{d}{dx}(\pi - 2x) = -2 $$ $$ \frac{d}{dx}(\tan x) = \sec^2 x $$ Now, we will find the limit of the ratio of the derivatives: $$ \lim_{x \rightarrow \frac{\pi}{2}} \frac{-2}{\sec^2 x} $$ As \(x\) approaches \(\frac{\pi}{2}\), the second squared becomes infinity: $$ \lim_{x \rightarrow \frac{\pi}{2}} \frac{-2}{\sec^2 x} = \frac{-2}{\infty} = 0 $$
06

Conclusion

The limit of the given expression as \(x\) approaches \(\frac{\pi}{2}\) is 0: $$ \lim_{x \rightarrow \frac{\pi}{2}}(\pi-2 x) \tan x = 0 $$

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