Evaluate the lim x to -infinity (3x^7 - x^6 + 7x)/(-5x^4 - 2x^2)

Hello Rames. In such a question, where x approaches infinity or -infinity, we should NOT factorize the given expression. But we should take the variable which has the highest power in the denominator, and we should divide using that variable in both the numerator and denominator.

In this question x⁴ is the variable with the highest power. So let's divide using it.

$\lim_{x \, \to \, -\infin}\frac{3x^7-x^6+7x}{-5x^4 - 2x^2}$

= $\lim_{x \, \to \, -\infin}\frac{\frac{3x^7}{x^4}-\frac{x^6}{x^4}+\frac{7x}{x^4}}{\frac{-5x^4}{x^4} - \frac{2x^2}{x^4}}$

= $\lim_{x \, \to \, -\infin} \frac{3x^3-x^2+\frac{7}{x^3}}{-5 - \frac{2}{x^2}}$

= $\frac{-\infin - \infin + 0^{-}}{-5 - 0^{-}}$

= ∞

I wrote $0^-$ for $\frac{7}{x^3}$ because if we keep the value of x as -999999 we get -7.000021000042001e-18 which is too small number approaching zero but still at the negative side.

We have ∞ value at the numerator, and we get -ve result in both the numerator and denominator. That's why ∞ is our answer.