limits question

Question

limit approaching infinity (-3x^5+3x^4-2/6x^5+5x^3+4x)

how do i go about this one? could i use substitution? factor then use substitution, or strictly graphing?

Answer 1

This is an example of a rational algebraic function in which degrees of numerator & denominator are identical.

 

In Algebra II, one learns that the END BEHAVIOR in such cases is y tends to a constant, equal to the ratio of leading coefficients.  Thus, in your case, this function has Horizontal Asymptote y = -½.

Answer 2

Divide the numerator and denominator by x⁵, the highest power of x in the denominator to get:

 

lim_x→∞[(-3 + 3/x - 2/x⁵)/(6 + 5/x² + 4/x⁴)] = (-3+0-0)/(6+0+0)

 

                                                            = -1/2