How do you evaluate the limit #(3x^4-x^2+5)/(10-2x^4)# as x approaches #oo#?

Question

Aurora Alvarado · Accepted Answer

# lim_(x rarr oo) (3x^4-x^2+5)/(10-2x^4) = -3/2# Note that as #x rarr oo# then #1/x,1/x^2,1/x^3 rarr 0# We can therefore multiply numerator and denominator by #1/x^4# (the reciprocal of the largest power in the denominator) as follows: # lim_(x rarr oo) (3x^4-x^2+5)/(10-2x^4) = lim_(x rarr oo) (3x^4-x^2+5)/(10-2x^4)*(1/x^4)/(1/x^4)# # " " = lim_(x rarr oo) ((1/x^4)(3x^4-x^2+5))/((1/x^4)(10-2x^4))# # " " = lim_(x rarr oo) (3-1/x^2+5/x^4)/(10/x^4-2)# # " " = (3-0+0)/(0-2)# # " " = -3/2# We can verify this result by looking at the graph of #y=(3x^4-x^2+5)/(10-2x^4)# graph{(3x^4-x^2+5)/(10-2x^4) [-10, 10, -5, 5]} and indeed it does appear that for large #x# the function is approaching a horizontal asymptote #y=-3/2#

Luke Alvarez · Answer

undefined To evaluate the limit as x approaches infinity, we can focus on the highest power terms in the numerator and denominator. In this case, the highest power terms are x^4 in both the numerator and denominator. By dividing each term by x^4, we can simplify the expression:

(3x^4 - x^2 + 5) / (10 - 2x^4)
= (3 - 1/x^2 + 5/x^4) / (10/x^4 - 2)

As x approaches infinity, 1/x^2 and 5/x^4 both approach zero, and 10/x^4 approaches zero as well. Therefore, the expression simplifies to:

(3 - 0 + 0) / (0 - 2)
= 3 / -2
= -3/2

Thus, the limit of (3x^4 - x^2 + 5) / (10 - 2x^4) as x approaches infinity is -3/2.