Given function f=4-3x^2/x^3, and domain ]1,infinity[, research if is asymptote to + infinity?

Expert Answers
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You need to find if the function has an oblique asymptote `y=mx+n`  when x tends to `+ oo` , hence, you need to find the values of m and n such that:

`m = lim_(x-gtoo) f(x)/x ; n = lim_(x-gtoo) (f(x) - m*x)`

Since n depends on m, you need to evaluate m first such that:

`lim_(x-gtoo) f(x)/x = lim_(x-gtoo) (4-3x^2)/x^4`

Forcing the factor `x^2`  to numerator yields:

`m = lim_(x-gtoo) (x^2(4/x^2-3))/x^4 = lim_(x-gtoo)(4/x^2-3)/x^2`

`m = -3/oo = 0`

You may evaluate n such that:

`n = lim_(x-gtoo) ((4-3x^2)/x^3 - 0*x)`

`n = 0`

Hence, evaluating if the function has an oblique asymptote when x tends to `+oo`  yields `y = 0` .