Limit (Calculus) Problem?

Question

Please help me figure out how to solve this. I already have the answer (-1/2), I just need assistance on what steps to take.

Limit [(1 / ( t * sqrt(1+t))) - (1 / t)]
t-> 0

Answer 1

Get the lcd of the expression

lim [1-√(1+t)] / [t√(1+t)]

then multiply by [1+√(1+t)] on both num & denom

lim [1-√(1+t)][1+√(1+t)] / [t√(1+t)][1+√(1+t)]
= lim -t / [t√(1+t)][1+√(1+t)]
= lim -1 / {√(1+t)[1+√(1+t)]}
= -1/2

Answer 2

Limit as t→0 { 1 / [t√(1 + t)] - 1/t }

Create a common fraction.

= Limit as t→0 { [1 - √(1 + t)] / [t√(1 + t)]

This is of the indeterminant form 0/0 so L'Hospital's Rule applies. The limit of the quotient is equal to the quotient of the derivative of the numerator divided by the derivative of the denominator.

= Limit as t→0 { (-1/[2√(1 + t)]) / (√(1 + t) + t/[2√(1 + t)]) }

= (-1/2) / (1 + 0) = -1/2

Answer 3

taking 1/t outside,
question becomes (1/t)[1/sqrt(1+t) -1 ]
=(1/t) [1-sqrt(1+t)]/sqrt(1+t)
multiplying numerator and denominator by (1+sqrt(1+t) )
(1-sqrt(1+t) )(1+sqrt(1+t) ) =1^2 - (sqrt(1+t))^2=1-1-t= -t
=(1/t) (-t/ [sqrt(1+t) * (1+sqrt(1+t) ) ]
cancelling t and putting t=0
= -1/[ 1 * (1+1) ]
= -1/2

Answer 4

take lcm
Limit [(1-sqrt(1+t) ) / (t*sqrt(1+t))]
t-> 0

introduce the conjugate (1+sqrt(1+t) ) in the numerator and denominator to get (-t) in the numerator

cancel off 't' and then apply limit