Given ( x + x^3 + x^5 ) / ( 1 + x^2 + x^4 ) how do you find the limit as x approaches negative infinity?

Jul 3, 2016

The quotient of polynomials such as this one 'behave' like the highest powers. In this case, this has the same behaviour as ${x}^{5} / {x}^{4} = x$

Explanation:

The limit of a sum is the sum of the limits, and the limit of quotient is the quotient of the limits, provided the individual limits exist. Also, the expression given doesn't change if we divide top and bottom by the same expression, as in:

$\frac{x + {x}^{3} + {x}^{5}}{x} ^ \frac{4}{1 + {x}^{2} + {x}^{4}} / {x}^{4}$, and distributing we get:

$\frac{\frac{x}{x} ^ 4 + {x}^{3} / {x}^{4} + {x}^{5} / {x}^{4}}{\frac{1}{x} ^ 4 + {x}^{2} / {x}^{4} + {x}^{4} / {x}^{4}} = \frac{\frac{1}{x} ^ 3 + \frac{1}{x} + x}{\frac{1}{x} ^ 4 + \frac{1}{x} ^ 2 + 1}$. So at the limit:

${\lim}_{x \rightarrow - \infty} \frac{\frac{1}{x} ^ 3 + \frac{1}{x} + x}{\frac{1}{x} ^ 4 + \frac{1}{x} ^ 2 + 1} = {\lim}_{x \rightarrow - \infty} \frac{0 + 0 + x}{0 + 0 + 1} = {\lim}_{x \rightarrow - \infty} x = - \infty$

As a general rule, dividing by the highest power in the denominator we can solve the problem.