How do you find the limit of #(2x^3+3x^2cosx)/(x+2)^3# as #x-oo#?

Question

How do you find the limit of #(2x^3+3x^2cosx)/(x+2)^3# as #x->oo#?

Luke Alvarez · Accepted Answer

The limit has indeterminate initial form #oo/oo#. We'll try to make the denominator go to someting oher than #oo#.

We want the limit as #xrarroo#, we aren't worried about what happens at #0#. For #x != 0#, #(2x^3+3x^2cosx)/(x+2)^3 = (x^3(2+(3cosx)/x))/(x(1+2/x))^3# # = (x^3(2+(3cosx)/x))/(x^3(1+2/x)^3)# # = (2+(3cosx)/x)/(1+2/x)^3# As #xrarroo#, the denominator goes to #1# and the numerator goes to #2# So the limit of the ratio as #xrarroo# is #2#

Emilia Aguilar · Answer

undefined To find the limit of (2x^3+3x^2cosx)/(x+2)^3 as x approaches infinity, we can use the concept of dominant terms.

First, we observe that as x approaches infinity, the cosine term oscillates between -1 and 1, and its effect becomes negligible compared to the polynomial terms.

Next, we focus on the dominant terms, which are the highest power terms in the numerator and denominator. In this case, it is 2x^3 in the numerator and (x+2)^3 in the denominator.

By dividing both the numerator and denominator by x^3, we can simplify the expression to (2 + 3cosx/x)/(1 + 2/x)^3.

As x approaches infinity, the cosine term becomes insignificant, and we are left with (2 + 0)/(1 + 0)^3, which simplifies to 2/1^3, or simply 2.

Therefore, the limit of (2x^3+3x^2cosx)/(x+2)^3 as x approaches infinity is 2.

How do you find the limit of #(2x^3+3x^2cosx)/(x+2)^3# as #x-&gt;oo#?

How do you find the limit of #(2x^3+3x^2cosx)/(x+2)^3# as #x->oo#?