L Hospital s Rule

Question

Evaluate lim (cos²(x))/x^3

x→0

Answer 1

L'Hospital's rule cannot be used in this case because cos²(x)/x³ with x approaching zero does not result in an indeterminate form (0/0, 0/∞, ∞/0, ∞/∞, etc).

In fact, if you consider cos(0) = 1, for numbers below 0, x³ will result in a negative (and as x approaches zero, x³ approaches -∞) while numbers above 0, x³ will result in a positive (and as x approaches zero, x³ approaches +∞) . Therefore, since the limit does not converge toward one number, the limit DNE.