How do you find the limit of # [sqrt (h^2 + 4h + 5) - sqrt(5)] / h # as h approaches 0?

1 Answer
Cesareo R.
Aug 1, 2016

#2/sqrt(5)#

Explanation:

# (sqrt (h^2 + 4h + 5) - sqrt(5)) / h = ((sqrt (h^2 + 4h + 5) - sqrt(5)) (sqrt (h^2 + 4h + 5) + sqrt(5)))/(h (sqrt (h^2 + 4h + 5) + sqrt(5)) ) = #
#=(h^2+4h+5-5)//(h (sqrt (h^2 + 4h + 5) + sqrt(5)) )=#
#=(h^2+4h)/(h (sqrt (h^2 + 4h + 5) + sqrt(5)) )=#
#=(h(h+4))/(h (sqrt (h^2 + 4h + 5) + sqrt(5)) )=#
#=(h+4)/((sqrt (h^2 + 4h + 5) + sqrt(5)) )=#

so

#lim_{h->0} (sqrt (h^2 + 4h + 5) - sqrt(5)) / h =lim_{h->0}(h+4)/((sqrt (h^2 + 4h + 5) + sqrt(5)) ) =2/sqrt(5)#

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