Question

`lim_(t->0)(1-sqrt(t/(t+1)))/(2-sqrt((4t+1)/(t+2))`?

Answer 1

Does not exist

Explanation:

first plug in 0 and you get (4+sqrt(2))/7
then test the limit on the left and right side of 0.
On the right side you get a number close to 1/(2-`sqrt(2)`)
on the left hand side you get a negative in the exponent which means the value doesn't exist.
The values on the left and right side of the function have to equal each other and they have to exist in order for the limit to exist.

Answer 2

`lim_(t->0)(1-sqrt(t/(t+1)))/(2-sqrt((4t+1)/(t+2))]=sqrt2/[2sqrt2-1]`

Explanation:

show below

`lim_(t->0)(1-sqrt(t/(t+1)))/(2-sqrt((4t+1)/(t+2))`

`=(1-sqrt0/(0+1))/(2-sqrt((4(0)+1)/(0+2))]=(1-0)/(2-sqrt((1)/(2))`

`(1)/(2-1/sqrt((2))]=sqrt2/[2sqrt2-1]`