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Chapter 4: Problem 44

Evaluate the following limits. $$\lim _{x \rightarrow \pi / 2} \frac{2 \tan x}{\sec ^{2} x}$$

Short Answer

Expert verified
Question: Evaluate the limit as \(x\) approaches \(\pi/2\) for the expression \(\frac{2\tan x}{\sec^2 x}\). Answer: The limit is 0 as \(x\) approaches \(\pi/2\).

Step by step solution

01

Recall trigonometric identities

We should recall that \(\tan x = \frac{\sin x}{\cos x}\) and \(\sec x = \frac{1}{\cos x}\). We will use these identities to rewrite the given expression in terms of \(\sin x\) and \(\cos x\).
02

Rewrite the expression using trigonometric identities

Substitute \(\tan x\) with \(\frac{\sin x}{\cos x}\) and \(\sec x\) with \(\frac{1}{\cos x}\) in the given expression: $$\frac{2 \tan x}{\sec ^{2} x} = \frac{2 \frac{\sin x}{\cos x}}{\left(\frac{1}{\cos x}\right)^{2}}$$
03

Simplify the expression

Now, simplify the expression: $$\frac{2 \frac{\sin x}{\cos x}}{\left(\frac{1}{\cos x}\right)^{2}} = \frac{2 \sin x \cos^2 x}{\cos x}$$
04

Cancel the common factors

We can cancel out one of the \(\cos x\) factors in the numerator and the denominator: $$\frac{2 \sin x \cos^2 x}{\cos x} = 2 \sin x \cos x$$
05

Evaluate the limit

Now evaluate the limit as \(x\) approaches \(\pi/2\): $$\lim _{x \rightarrow \pi / 2} \left(2 \sin x \cos x\right)$$ As \(x\) approaches \(\pi/2\), \(\sin x\) approaches 1, and \(\cos x\) approaches 0: $$\lim _{x \rightarrow \pi / 2} \left(2 \sin x \cos x\right) = 2 \cdot 1 \cdot 0 = 0$$ So, the limit is 0.

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