evaluate the limit: lim x approaches 0 (sqr 1 + t) - sqr 1 - t / t

Question

_(√1 + t)_-_(√1-t)_

t

**note: there are no parenthesis in the equation!

Answer 1

lin as t→0 of [(sqrt(1 + t) - sqrt(1 -t)]/t. This is indeterminate form (0/0). We multiply numerator and denominator by congugate to numerator expression:

lin as t→0 of [(sqrt(1 + t) - sqrt(1 -t)]/tlin as t→0 of 2/[sqrt(1 + t) + sqrt(1 - t)] = 2/2 = 1

Answer 2

lim _t-->0 (√(1 + t) -√(1 - t) )/t

=|multiply both numerator ad denominator by (√(1 + t)+√(1 + t) )| =

=lim _t-->0 (1 + t -1 +t) / (t(√(1 + t) +√(1 + t) ) ) =

=lim _t-->0 (2t) / (t(√(1 + t) +√(1 + t) ) =

=2 lim _t-->0 1 / (√(1 + t) +√(1 - t) ) = 2*1/2=1

Answer 3

Indeterminant 0/0

derivative of t in denominator is 1

L'Hopitals:

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