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  • <img alt="\mathrm{Evaluate \lim _{t \rightarrow-\infty} \frac{2-t+\sin t}{t+\cos t} }" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7BEvaluate%20%5Clim%20_%7Bt%20%5Crightarrow-%5C

\mathrm{Evaluate \lim _{t \rightarrow-\infty} \frac{2-t+\sin t}{t+\cos t} }

Option: 1

-1


Option: 2

3


Option: 3

0


Option: 4

2


Answers (1)

best_answer

\mathrm{ \lim _{t \rightarrow \infty} \frac{2-t+\sin t}{t+\cos t}=\lim _{t \rightarrow \infty} \frac{t\left(\frac{2}{t}-1+\frac{\sin t}{t}\right)}{t\left(1+\frac{\cos t}{t}\right)}=-1 }

since it is sufficient to observe that:

1) \mathrm{\frac{2}{t} \rightarrow 0.}

2) \mathrm{\frac{\sin t}{t}} and  \mathrm{\frac{\cos t}{t}}go to 0 since are both of them the product between a function that goes to \mathrm{0\left(\frac{1}{t}\right)} and a bounded

function \mathrm{( \cos t \, and \sin t ).}

Posted by

HARSH KANKARIA

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