What is the limit of # (sqrt(x^4 - 6x^2)) - x^2# as x goes to infinity?

1 Answer
Jim H
Oct 19, 2015

#lim_(xrarroo) (sqrt(x^4 - 6x^2) - x^2) = -3#

Explanation:

#lim_(xrarroo) (sqrt(x^4 - 6x^2) - x^2)# has indeterminate form #oo-oo#.

So we'll do some algebra:

# sqrt(x^4 - 6x^2) - x^2 = ((sqrt(x^4 - 6x^2) - x^2))/1 * ((sqrt(x^4 - 6x^2) + x^2)) / ((sqrt(x^4 - 6x^2) + x^2)) #

# = ((x^4-6x^2)-x^4)/(sqrt(x^4 - 6x^2) + x^2) # #" "# (has indeterminate form #oo/oo#)

# = (-6x^2)/(sqrt(x^4(1-6/x^2))+x^2)# for #x != 0#

# = (-6x^2)/(sqrt(x^4)sqrt((1-6/x^2))+x^2)# for #x != 0#

# = (-6x^2)/(x^2(sqrt(1-6/x^2)+1))# for #x != 0#

# = (-6)/(sqrt(1-6/x^2)+1)# for #x != 0#

Now as #xrarroo#, we see that the ratio #rarr (-6)/2 #

So, #lim_(xrarroo) (sqrt(x^4 - 6x^2) - x^2) = -3#

Note that because the identity #sqrt(x^4)=x^2# is true for both positive and negative values of #x#, we also have

#lim_(xrarr-oo) (sqrt(x^4 - 6x^2) - x^2) = -3#

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