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lim(x rarr 3) (sqrt(5x+1)-sqrt(7x-5))/(s...

`lim_(x rarr 3) (sqrt(5x+1)-sqrt(7x-5))/(sqrt(7x+4)-sqrt(5x+10))=`_____.

A

0

B

`-(5)/(4)`

C

`(5)/(4)`

D

`oo`

Text Solution

Verified by Experts

The correct Answer is:
B
`underset(x rarr 3)("lim")(sqrt(5x+1)-sqrt(7x-5))/(sqrt(7x+4)-sqrt(5x+10))=(0)/(0)`
That is, indeterminate form.
Multiply both the numerator and the denominator with `(sqrt(5x+1)+sqrt( 7x-5))(sqrt(7x+4)+sqrt(5x+10)`
`rArr underset(x rarr 3)("lim")([5x+1-(7x-5)](sqrt7x+4)+sqrt(5x+10))/((7x+4-5x-10)(sqrt(5x+1)+sqrt(7x-5)))`
`rArr underset(x rarr 3)("lim")((-2x+6)(sqrt(7x+4)+sqrt(5x+10)))/((2x-6)(sqrt(5x+1)+sqrt(7x-5)))`
`=underset(x rarr 3)("lim")(-(sqrt(7x+4)+sqrt(5x+10)))/((sqrt(5x+1)+sqrt(7x-5)))`
`=(-(sqrt(25)+sqrt(25)))/((sqrt(16)+sqrt(16)))=(-10)/(8)=(-5)/(4)`
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