Lim without using de l'hospital method.. could you help?

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How to resolve it without using de l'hospital method?

1 Answer
Cesareo R.
Dec 12, 2017

#-3/8#

Explanation:

Making #y= x-2# we have

#lim_(x->2)(sqrt(3x+10)-4)/(1-e^(x-2)) = lim_(y->0)(sqrt(3y+16)-4)/(1-e^y)=lim_(y->0)(3y)/((1-e^y)(sqrt(3y+16)+4))#

but for small #absy# we have #e^y = 1 + y +O(y^2)# then

#lim_(x->2)(sqrt(3x+10)-4)/(1-e^(x-2)) = lim_(y->0)(3y)/((-y+O(y^2))(sqrt(3y+16)+4)) = -3/8#

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