lim 1/√ (x-1) - 1/√ (x^2 -1)
x--> 0
x--> 0
-
lim_(x->0) (1/sqrt(x-1)-x/sqrt(x^2-1))
The limit of a difference is the difference of the limits:
= lim_(x->0) 1/sqrt(x-1)-lim_(x->0) x/sqrt(x^2-1)
The limit of a quotient is the quotient of the limits:
= 1/(lim_(x->0) sqrt(x-1))-lim_(x->0) x/sqrt(x^2-1)
Using the power law, write lim_(x->0) sqrt(x-1) as sqrt(lim_(x->0) (x-1)):
= 1/sqrt(lim_(x->0) (x-1))-lim_(x->0) x/sqrt(x^2-1)
The limit of x-1 as x approaches 0 is -1:
= -(lim_(x->0) x/sqrt(x^2-1))-i
The limit of a quotient is the quotient of the limits:
= -(lim_(x->0) x)/(lim_(x->0) sqrt(x^2-1))-i
The limit of x as x approaches 0 is 0:
= -i
The limit of a difference is the difference of the limits:
= lim_(x->0) 1/sqrt(x-1)-lim_(x->0) x/sqrt(x^2-1)
The limit of a quotient is the quotient of the limits:
= 1/(lim_(x->0) sqrt(x-1))-lim_(x->0) x/sqrt(x^2-1)
Using the power law, write lim_(x->0) sqrt(x-1) as sqrt(lim_(x->0) (x-1)):
= 1/sqrt(lim_(x->0) (x-1))-lim_(x->0) x/sqrt(x^2-1)
The limit of x-1 as x approaches 0 is -1:
= -(lim_(x->0) x/sqrt(x^2-1))-i
The limit of a quotient is the quotient of the limits:
= -(lim_(x->0) x)/(lim_(x->0) sqrt(x^2-1))-i
The limit of x as x approaches 0 is 0:
= -i