Evaluate the following limit:lim x→∞ [√(2x2-3)-√(2) x]

suchita undare
20 Points
12 years ago

the above function takes ∞-∞ indeterminant form

rationalising it we get

lim x→∞ [√(2x2-3)-√(2) x]*[√(2x2-3)+√(2) x]/[√(2x2-3)+√(2) x]

=lim x→∞ -3/[√(2x2-3)+√(2) x]

this is not an indeterminant form so put x=∞ in the expression

=-3/(∞+∞)

=0

hope this helps

290 Points
12 years ago

Hi Menka,

lim x→∞  x[√(2-3/x2) - √2]

lim x→∞  x [(2-3/x2 - 2)/(√(2-3/x2)+√2)] ------------- {Multiplying Nr and Dr by the conjugate of the Nr}

lim x→∞  -3/[(x)*(√(2-3/x2)+√2)].

Now this, as x→∞, will tend to 0. {As x is in the Dr}.

Hence the limit of the question is 0.

Hope that helps.

All the Best,

Regards,

Ashwin.

RAJORSHI PAUL
36 Points
12 years ago

lim x->inf (2x^2-3)^0.5-2^0.5x

Replace x by 1/y such that y ->0

so lim y->0 (1/y){(2-3y^2)^0.5-2^0.5}[0/0 form]

=lim y->0 (-3)y/(2-3y^2)^0.5[L Hospital's rule] =0

MATH SOLVER
18 Points
12 years ago

we never discuss ∞+∞ forms.

The answer is Wrong for 2 reasons

1) The given limit is in the form ∞-∞ which cannot be solved

2) Look for the image below

290 Points
12 years ago

Hi Math Solver,

Kindly note that the denominator as x→∞, will be of the form ∞*(√2+√2).....

Qnd hence the limiting value is zero....

Best Regards,