L’Hopital’s Rule is a powerful tool for evaluating limits involving Indeterminate Forms.
L’Hopital’s Rule Statement
If the limit of a fraction as $ x $ approaches $ a $ results in an indeterminate form (0/0 or ∞/∞), then:
$$ \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f’(x)}{g’(x)} $$
provided that the limit on the right-hand side exists or is $ \pm \infty $ .
In simpler terms: To evaluate the limit of a fraction that results in an indeterminate form, take the derivative of both the numerator and denominator separately, and then evaluate the limit again.
Important Notes
-
L’Hopital’s Rule only applies to Indeterminate Forms (0/0 or ∞/∞). Don’t use it for other forms like 0/∞ or ∞/0.
- The rule can be applied repeatedly if the limit of the derivatives still results in an indeterminate form.
- L’Hopital’s Rule is not always the easiest method to evaluate a limit. Sometimes, algebraic manipulation or other techniques may be more efficient.
Example
Let’s say we want to evaluate the limit:
$$ \lim_{x \to 0} \frac{\sin(x)}{x} $$ 1. Indeterminate Form: Substituting $ x = 0 $ directly gives us $ \frac{\sin(0)}{0} = \frac{0}{0} $ , which is an indeterminate form.
2. Apply L’Hopital’s Rule:
- $ f(x) = \sin(x) $
- $ g(x) = x $
- $ f’(x) = \cos(x) $
- $ g’(x) = 1 $
Therefore, we have:
$$ \lim_{x \to 0} \frac{\sin(x)}{x} = \lim_{x \to 0} \frac{\cos(x)}{1} $$
3. Evaluate the Limit:
$$ \lim_{x \to 0} \frac{\cos(x)}{1} = \frac{\cos(0)}{1} = 1 $$
So, $ \lim_{x \to 0} \frac{\sin(x)}{x} = 1 $ .