Evaluate the limit, using L'Hopital's Rule if necessary

Question

lim e^{x^2}/1-x³

x-->infinity

Answer 1

Note that by plugging in x=∞ , the function will be of the form ∞/∞ which is indeterminate, and we can use L'Hopital's rule.

lim e^{x^2}/(1-x³ ) as x goes to ∞

= lim (e ^{x^2} * 2x) / (-3x²) , by taking the derivative of numerator (apply chain rule) and denominator, via L'Hopital's rule

= lim 2 e ^x*2/(-3x) after simplifying, canceling out the common factor x

Note that the above equation is in the form ∞/∞ when you plug in x=∞

=lim 2ex^2 *(2x) / (-3) , by taking the derivative of numerator and denominator via L'Hopital's rule

= - ∞ (negative infinity)

Answer 2

1. Divide numerator and denominator by highest power, x³
2. (e^{x^2}/x³)
3. (1-x³)/x³
4. Evaluate each limit as x approaches infinity
5. As x approaches infinity, (e^{x^2}/x³) = infinity
6. As x approaches infinity, (1-x³)/x³ = -1
7. infinity/-1 = -infinity