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Question Number 77182 by jagoll last updated on 04/Jan/20

$$$$$$$$$$\mathrm{evaluate}\underset{x\to 0}{\lim }\frac{\underset{\mathrm{a}}{\stackrel{\mathrm{x}}{\int }}\left(\frac{\cos \mathrm{t}}{\mathrm{t}}\right)\mathrm{dt}}{\mathrm{x}}.$$

Commented by mathmax by abdo last updated on 05/Jan/20

$$ithinkthata=0...$$

Answered by john santu last updated on 04/Jan/20

$$supposeF\left(t\right)isantiderivative$$$$\int \frac{\cos t}{t}dtsuchthat\frac{dF\left(t\right)}{dt}=\frac{\cos \left(t\right)}{t}.$$$$now\underset{x\to 0}{\lim }\frac{F\left(t\right)\underset{a}{\stackrel{x}{\mid }}}{x}=\underset{x\to 0}{\lim }\frac{F\left(x\right)-F\left(a\right)}{x}$$$$weassumethatF\left(0\right)-F\left(a\right)=0$$$$\underset{x\to 0}{\lim }\frac{F^{\prime }\left(x\right)-F^{\prime }\left(a\right)}{1}=F^{\prime }\left(0\right)-F^{\prime }\left(a\right)$$$$but{F}^{\prime }\left(0\right)=\frac{\cos \left(0\right)}{0}undefined$$$$ithinkitproblemerror.$$$$$$

Commented by jagoll last updated on 04/Jan/20

$$\mathrm{thanks}\mathrm{sir}.\mathrm{i}\mathrm{will}\mathrm{check}\mathrm{my}\mathrm{question}$$

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