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Chapter 9: Problem 727

\(\lim _{\mathrm{X} \rightarrow 0}(\pi)^{(\mathrm{e}) \mathrm{x}}(\mathrm{e})^{(-\pi) \mathrm{x}}=?\) (a) \((\mathrm{e} / \pi)\) (b) \((\pi / \mathrm{e})\) (c) 1 (d) Limit does not exist

Short Answer

Expert verified
The short answer to the question is: \[\lim_{x \to 0}(\pi)^{(e)x}(e)^{(-\pi)x} = 1\]

Step by step solution

01

Rewrite the expression in a simplified form

First, let's rewrite the given expression in a general form. The given expression is \((\pi)^{(e)x}(e)^{(-\pi)x}\). Using the property of exponentials, \((a^x)(b^x)= (ab)^x\), we can rewrite the expression as follows: \[((\pi^e)(e^{-\pi}))^x\]
02

Set x to 0

Now that we have a simplified form, we need to find the limit as x approaches 0 of this expression. We can use the fact that any value raised to the power of 0 is 1: \[a^0 = 1\] So, when x = 0, our expression becomes: \[((\pi^e)(e^{-\pi}))^0\]
03

Calculate the limit of the simplified expression

Since we have the expression in the form of \(((\pi^e)(e^{-\pi}))^0\), we know that \[((\pi^e)(e^{-\pi}))^0 = 1^0\]. As previously stated, any value raised to the power of 0 is 1. So the limit of the expression as x approaches 0 is: \[\lim_{x \to 0}((\pi^e)(e^{-\pi}))^x = 1\]
04

Check the options

Finally, we find the option that matches our calculated limit: (a) \((\frac{e}{\pi})\) (b) \((\frac{\pi}{e})\) (c) 1 (d) Limit does not exist The option (c) 1 matches our calculated limit. So, the solution is: \[\lim_{x \to 0}(\pi)^{(e)x}(e)^{(-\pi)x} = 1\]

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Most popular questions from this chapter

\(\lim _{\mathrm{x} \rightarrow 0}\left[\left\\{\tan ^{108}(107 \mathrm{x})\right\\} /\left\\{\log \left(1+\mathrm{x}^{108}\right)\right\\}\right]=?\) (a) \((107 / 108)\) (b) \((107)^{108}\) (c) \((107)^{-108}\) (d) \(-(107 / 108)\)

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If \(=\mid \begin{array}{ll}\mathrm{x}+\mathrm{a} \sqrt{2} \operatorname{Sin} \mathrm{x}, & 0 \leq \mathrm{x} \leq(\pi / 4) \\ 2 \mathrm{x} \operatorname{Cot} \mathrm{x}+\mathrm{b}, & {[(\pi / 4)<\mathrm{x} \leq(\pi / 2)]} \\ \mathrm{a} \operatorname{Cos} 2 \mathrm{x}+\mathrm{b} \operatorname{Sin} \mathrm{x}, & (\pi / 2)<\mathrm{x} \leq \pi\end{array}\) is continuous on \([0, \pi]\), then \(\mathrm{a}=\ldots \ldots\) and \(\mathrm{b}=\ldots \ldots .\) (a) \(\mathrm{a}=(5 \pi / 2), \mathrm{b}=(5 \pi / 4)\) (b) \(\mathrm{a}=-(5 \pi / 2), \mathrm{b}=-(5 \pi / 4)\) (c) \(\mathrm{a}=(\pi / 6), \mathrm{b}=[(-\pi) / 12]\) (d) \(\mathrm{a}=-(5 \pi / 4), \mathrm{b}=(5 \pi / 2)\)

If \(\lim _{\mathrm{x} \rightarrow 0}[\\{\operatorname{Sin}[(\mathrm{n}+1) \mathrm{x}]+\sin \mathrm{x}\\} / \mathrm{x}]=(1 / 2)\) then value of n is: (a) \(-2.5\) (b) \(-0.5\) (c) \(-1.5\) (d) \(-1\)

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