How do you evaluate the limit #(x^4+3^x)/(x^5+1)# as x approaches #oo#?

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Answer 1 · 2016-08-29T10:00:02

#oo#

#lim_(x to oo) (x^4+3^x)/(x^5+1)#

this is in #oo/oo# indeterminate form and therefore L'Hopital's rule applies on repeated basis as follows

#= lim_(x to oo) (4x^3+ln (3) 3^x)/(5x^4)#

#= lim_(x to oo) (12x^2+ln^2 (3) 3^x)/(20x^3)#

#= lim_(x to oo) (24x+ln^3 (3) 3^x)/(60x^2)#

#= lim_(x to oo) (24+ln^4 (3) 3^x)/(120x)#

#= lim_(x to oo) (ln^5 (3) 3^x)/(120)#

#= infty#