WHEN TO USE L'HOPITAL'S RULE?

Question

1. For different types of rational functions (you can use rational expressions where both numerator and denominator are polynomials), what methods would you recommend following aside from  L'HÔPITAL'S RULE? Provide examples.

1. After solving this rational function consider that this time, you use both L’Hopital’s rule and the previous method (algebra of limits, Sandwich Theorem, etc.). Which method is more convenient to use?

Answer 1

The other situation in which rational fractions misbehave is at a zero of the denominator.

When the numerator is non-zero and the denominator is 0, the fraction is undefined and has a non-removable discontinuity.

The tricky situation is when both numerator and denominator are zero at the limit. Here, l'Hospital's rule will work but it is usually easier to find the factor common to the numerator and denominator and divide it out.

Example:

lim x->1 of (x² + x -2)/((x² - 1) = lim x->1 (x+2)/(x+1) = 3/2.

Answer 2

If you have a fraction of polynomials and you are taking the limit as x goes to infinity, you can just look at the ratio of the leading term. If the leading terms are not the same order, the larger order term dominates. If they are the same order, than the limit is the ratio of the coefficients of the leading terms. L'Hopital's Rule works, but it is unnecessary to take the derivative repeatedly as all the terms below the leading terms will disappear as you keep taking the derivative. Here are examples all of them are limits as x ⇒ ∞

Applying L'Hopital's

(x²+1)/x → 2x/1 → ∞ Obviously the inverse fraction would go to 1/2x or 0

(x²+1)/(3x²+2x-15) → (2x)/(6x+2) → 2/6 →1/3

All of these could be obtained much faster just using the limiting behavior of the leading terms.