Question

Find the limit. Use l'Hospital's Rule if appropriate. If there is a more elementary method, consider using it.

lim (x + x²)/(9 − 5x²)
x→[infinity]

Answer 1

To find the limit of (x + x²)/(9 - 5x²) as x approaches infinity, we can divide both the numerator and denominator by the highest power of x, which is x²:

(x + x²)/(9 - 5x²) = (x²(1 + 1/x))/(x²(-5/ x² + 9/ x²))

Simplifying, we get:

(x²(1 + 1/x))/(x²(-5/ x² + 9/ x²)) = (1 + 1/x)/(-5/ x² + 9/ x²)

As x approaches infinity, both -5/x² and 9/x² approach zero, so we can evaluate the limit as:

lim (1 + 1/x)/(-5/ x² + 9/ x²) = lim (1 + 1/x)/(4/ x²)

Now we can apply l'Hospital's rule, taking the derivative of the numerator and denominator with respect to x:

lim (1 + 1/x)/(4/ x²) = lim (-1/x²)/(8/ x³) = lim (-x)/(8) = -∞

Therefore, the limit of (x + x²)/(9 - 5x²) as x approaches infinity is -∞.