How do you evaluate the limit #(sqrt(4+h)-2)/h# as h approaches #0#?

Question

14263 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-08-14T08:12:39

# = 1/4#

#lim_(h to 0) (sqrt(4+h)-2)/h#

setting it up for a binomial expansion
#= lim_(h to 0) 1/h( 2(1+h/4)^(1/2)-2)#

Expanding...
#= lim_(h to 0) 1/h( 2 ( 1+(1/2) h/4 + O(h^2))-2)#

#= lim_(h to 0) 1/h( 2 + h/4 + O(h^2)-2)#

#= lim_(h to 0) 1/h( h/4 + O(h^2))#

#= lim_(h to 0) 1/4 + O(h)#

# = 1/4#

you could use L'Hopital here too