Let f(x)=|8x−6x3|/(x2+1)(3x−8). Find lim x→∞ f(x).

Question

Let f(x)=|8x−6x3|/(x²+1)(3x−8).

Find lim x→∞ f(x).

Answer 1

 f(x)=|8x−6x³|/(x²+1)(3x−8) = |6x³− 8x|/(x²+1)(3x−8).

We want to know what is the value of y (or what value does y approach when x gets larger and larger).

To find the limit as x approaches infinity of a rational function, we consider the ratio of the degree term of the numerator to the degree term of the denominator.

The degree term of the numerator = 6x³ (note that the absolute value makes it positive).

The degree term of the denominator = (x²)(3x) = 3x³

Hence,  lim x→∞ f(x) = lim x→∞ 6x³/3x³ = 2

The more correct process to evaluate this limit is to divide every term in the rational function by x³, and note that lim x→∞ 1/x = 0. This will leave you with 6/3 = 2.