Question #60777

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Answer 1 · 2017-01-15T21:02:57

#lim_(xrarroo)(10x^2+7x-3)/sqrt(5x^4+2x^2-5x)=2sqrt5#

Factor the largest degree of #x# from the numerator and denominator.

#=lim_(xrarroo)(x^2(10+7/x-3/x^2))/sqrt(x^4(5+2/x^2-5/x^3))#

Pulling out #sqrt(x^4)# as #x^2#:

#=lim_(xrarroo)(x^2(10+7/x-3/x^2))/(x^2sqrt(5+2/x^2-5/x^3))#

These cancel:

#=lim_(xrarroo)(10+7/x-3/x^2)/(sqrt(5+2/x^2-5/x^3))#

As #x# goes to infinity, any term like #3/x# or #-5/x^3# will approach #0#:

#=(10+7/oo-3/oo)/sqrt(5+2/oo-5/oo)#

#=(10+0+0)/sqrt(5+0+0)#

#=10/sqrt5#

#=(10sqrt5)/5#

#=2sqrt5#