How do you find the limit #(sqrt(x^2+1)-1)/(sqrt(x+1)-1)# as #x->0#?

1 Answer
Jim H
Dec 13, 2016

Use a variation of the same method used for similar limits.

Explanation:

The initial form is #0/0#. Changing just one of the subtractions to addition won't work, so change both.

#((sqrt(x^2+1)-1))/((sqrt(x+1)-1)) * ((sqrt(x^2+1)+1)(sqrt(x+1)+1))/((sqrt(x^2+1)+1)(sqrt(x+1)+1)) = (((x^2+1)-1)(sqrt(x+1)+1))/(((x+1)-1)(sqrt(x^2+1)+1)#

# = (x^2(sqrt(x+1)+1))/(x(sqrt(x^2+1)+1))#

# = (x(sqrt(x+1)+1))/(sqrt(x^2+1)+1)#

Taking the limit as #xrarr0# yields

# = ((0)(1+1))/((1)+1) = 0#

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