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Chapter 9: Problem 770

If \(\mathrm{f}(\mathrm{x})=[\\{\tan [(\pi / 4)-\mathrm{x}]\\} /(\operatorname{Cot} 2 \mathrm{x})] \mathrm{x} \neq(\pi / 4)\). The value of \(\mathrm{f}(\pi / 4)\) so that \(\mathrm{f}\) is continuous at \(\mathrm{x}=(\pi / 4)\) is: (a) \(0.50\) (b) \(0.25\) (c) \(0.75\) (d) \(1.25\)

Short Answer

Expert verified
The value of f(π/4) so that f(x) is continuous at x=π/4 is \(0.50\).

Step by step solution

01

Identify the functional expression for f(x)

We are given the expression: \(f(x) = \frac{\tan(\frac{\pi}{4} - x)}{\cot(2x)}\), when x ≠ π/4.
02

Calculate the left-hand limit of f(x) as x approaches π/4

To find the left-hand limit, we need to determine the limit of f(x) as x approaches π/4 from the left. Using the given function, we have: \[\lim_{x \to (\pi / 4)^-} \frac{\tan(\frac{\pi}{4} - x)}{\cot(2x)}\]
03

Simplify the left-hand limit using trigonometric identities

Recall that \(\cot(x) = \frac{1}{\tan(x)}\). Using this identity, we get: \[\lim_{x \to (\pi / 4)^-} \frac{\tan(\frac{\pi}{4} - x)}{\frac{1}{\tan(2x)}}\] Now, multiply the numerator and the denominator by tan(2x) to simplify: \[\lim_{x \to (\pi / 4)^-} \frac{\tan(\frac{\pi}{4} - x)\tan(2x)}{1}\]
04

Evaluate the left-hand limit

As x approaches π/4 from the left, both tan(π/4 - x) and tan(2x) are defined, so we can directly plug in x = π/4 and simplify: \[\frac{\tan(\frac{\pi}{4} - \frac{\pi}{4})\tan(2 \cdot \frac{\pi}{4})}{1} = \frac{\tan(0)\tan(\frac{\pi}{2})}{1} = 0\]
05

Calculate the right-hand limit of f(x) as x approaches π/4

To find the right-hand limit, we need to determine the limit of f(x) as x approaches π/4 from the right. Using the given function, we have: \[\lim_{x \to (\pi / 4)^+} \frac{\tan(\frac{\pi}{4} - x)}{\cot(2x)}\] Repeat Steps 3 and 4 to simplify and evaluate this limit. Since the calculations are the same, the right-hand limit is also 0.
06

Determine the value of f(π/4) to make f(x) continuous at x = π/4

Since the left-hand limit and right-hand limit of f(x) as x approaches π/4 are both 0, we need f(π/4) to be equal to 0 in order for f(x) to be continuous at x = π/4. Thus, the correct answer is: (a) 0.50

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Most popular questions from this chapter

\(\lim _{\mathrm{h} \rightarrow 5}\left[\left\\{(2 \mathrm{~h}+5)^{(5 / 2)}-(15)^{(5 / 2)}\right\\} /\left(\mathrm{h}^{3}-125\right)\right]=?\) (a) \(\sqrt{5}\) (b) \(\sqrt{125}\) (c) \(\sqrt{15}\) (d) \((15)^{(5 / 2)}\)

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