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The value of the limit lim(x to -oo) (sq...

The value of the limit `lim_(x to -oo) (sqrt(4x^(2) - x + 2x))` is

A

`-oo`

B

`(-1)/(4)`

C

`0`

D

`(1)/(4)`

Text Solution

Verified by Experts

The correct Answer is:
D
Given form is `(rarr oo - rarr oo)` reverse rationalizing the expression to convert it to `(rarr oo)/(rarr oo)`
`sqrt(4x^(2)-x)+2x xx ((sqrt(4x^(2)-x)-2x))/((sqrt(4x^(2)-x)-2x))`
`=(4x^(2)-x-4x^(2))/(sqrt(4x^(2)-x)-2x)=(-x)/(sqrt(4x^(2)-x)-2x)implies (-x)/(|x|sqrt(4-(1)/(x)-2x))("Taking" sqrt(x^(2)=)|x|)`
Now as `x rarr - oo |x| = - x ` thus the expression becomes
`implies (-x)/(-xsqrt(4-(1)/(x)-2x))=(1)/(sqrt(4-(1)/(x)+2)) "put" x = - oo = (1)/(2+2) =(1)/(4)`
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