How do you evaluate # ( x^2+2) /( sqrt (9x^4 +1))# as x approaches infinity?

1 Answer
Jim H
Apr 19, 2016

Rewrite using algebra so that the limit does not have indeterminate form.

Explanation:

#lim_(xrarroo)(x^2+2)/sqrt(9x^4+1)# has initial form #oo/oo# which is an indeterminate form.

#(x^2+2)/sqrt(9x^4+1) = (x^2+2)/sqrt(x^4(9+1/x^4)# #" "# for all #x!= 0#

# = (x^2+2)/(sqrt(x^4)sqrt(9+1/x^4))# #" "# for all #x!= 0#

# = (x^2+2)/(x^2sqrt(9+1/x^4))# #" "# for all #x!= 0#

# = (x^2(1+2/x^2))/(x^2sqrt(9+1/x^4))# #" "# for all #x!= 0# since #sqrt(x^4) = x^2#

# = (1+2/x^2)/sqrt(9+1/x^4)# #" "# for all #x!= 0#

And

#lim_(xrarroo) (1+2/x^2)/sqrt(9+1/x^4) = 1/sqrt9 = 1/3#

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