Calculus: Early Transcendentals, Chapter 2, 2.3, Section 2.3, Problem 29

lim_(t->0) (1/(tsqrt(1+t)) - 1/t)
sol:
lim_(t->0) (1/(tsqrt(1+t)) - 1/t)
=> lim_(t->0) ((1- sqrt(1+t))/(tsqrt(1+t)) )
now simplify the numerator
1- sqrt(1+t) = (1- sqrt(1+t)) *((1+ sqrt(1+t))/(1+ sqrt(1+t)))
= (1^2- (sqrt(1+t))^2)/(1+ sqrt(1+t))
so,
lim_(t->0) ((1-sqrt(1+t))/(tsqrt(1+t)) )
=lim_(t->0)((1^2- (sqrt(1+t))^2)/(1+ sqrt(1+t)))/((tsqrt(1+t)))
= lim_(t->0) (-t/((1+ sqrt(1+t))(t(sqrt(1+t)))))
= lim_(t->0)(-1/((1+ sqrt(1+t))(sqrt(1+t))))
=(lim_(t->0)(-1))/ (lim_(t->0)((1+ sqrt(1+t))(sqrt(1+t)))) --------(1)
as,the denominator is
(lim_(t->0)(1+ sqrt(1+t)(sqrt(1+t))))
=(lim_(t->0)(1+ sqrt(1+t)) (lim_(t->0)(sqrt(1+t)))
as t-> 0 we get
(lim_(t->0)(1+ sqrt(1+t)) = (1+1) = 2
(lim_(t->0)(sqrt(1+t))) = 1
so,
(lim_(t->0)(1+ sqrt(1+t)) (lim_(t->0)(sqrt(1+t)))= (2) (1) =2
so, from (1)
we get
lim_(t->0)(-1)/ (lim_(t->0)(1+ sqrt(1+t)(sqrt(1+t)))) = -1 /2
there fore
lim_(t->0) (1/(tsqrt(1+t)) - 1/t) = -1/2