Evaluate the limit, if it exists

Question

lim (t-->0)  (√(1 + t) - √(1 - t))/t

 

ANSWER: 1

#25, p107

Answer 1

Multiply top and bottom of the original by the conjugate of the numerator. The conjugate is √(1+t) + √(1-t).

 

This gives:

 

(1+t) - (1-t) in the numerator (multiply sum/difference of same expression gives a difference of squares)

 

In the denominator we get:

 

t[√(1+t) + √(1-t)].

 

The numerator simplifies to 2t. Now the t in the numerator and denominator cancel so we have:

 

lim_t->0 2/[√(1+t) + √(1-t)] = 2/[1+1] = 1.