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lim(x->-oo)[x^5tan(1/[pix^2])+3|x|^2+7]/...

`lim_(x->-oo)[x^5tan(1/[pix^2])+3|x|^2+7]/[|x|^3+7|x|+8]`

Text Solution

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`Lim_(x->-oo) (x^5tan(1/(pix^2))+3|x|^2+7)/(|x|^3+7|x|+8)`
As, this is a `oo/oo` form. So, we will apply L`'`Hospital rule.
`=Lim_(x->-oo) (5x^4tan(1/(pix^2))+x^5(sec^2(1/(pix^2)))(-2/(pix^3))+6|x|)/(3|x|^2 +7)`
`=Lim_(x->-oo)((x^2(5x^2tan(1/(pix^2))-2/pi sec^2(1/(pix^2))-6/x))/(x^2(-3+7/x)))`
`=Lim_(x->-oo)(5x^2tan(1/(pix^2))-2/pi sec^2(1/(pix^2))-6/x)/(-3+7/x)`
`=(0-2/pi(1)-0)/(-3+0)`
`=2/(3pi)`
`:. Lim_(x->-oo) (x^5tan(1/(pix^2))+3|x|^2+7)/(|x|^3+7|x|+8) = 2/(3pi)`
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