L'Hôpital's Rule

Definition

L'Hôpital's Rule is a mathematical method used to evaluate limits that result in indeterminate forms, typically when direct substitution yields results like 0/0 or ∞/∞. This rule states that for functions f(x) and g(x) that are differentiable in a neighborhood around a point (except possibly at the point itself), if both f(x) and g(x) approach 0 or ±∞, the limit of their quotient can be found by taking the limit of the derivatives of these functions instead.

5 Must Know Facts For Your Next Test

1. L'Hôpital's Rule applies only to certain indeterminate forms, most commonly 0/0 and ∞/∞, but can also be used for other forms like 0·∞, ∞ - ∞, 0^0, ∞^0, and 1^∞ by appropriate manipulation.
2. The rule states that if $$rac{f(x)}{g(x)}$$ gives an indeterminate form, then $$ ext{lim}_{x o c} rac{f(x)}{g(x)} = ext{lim}_{x o c} rac{f'(x)}{g'(x)}$$ if the limit on the right side exists.
3. It is essential that both f(x) and g(x) are differentiable in an interval around c, excluding possibly at c itself, for L'Hôpital's Rule to be valid.
4. If applying L'Hôpital's Rule results in another indeterminate form, it can be applied repeatedly until a determinate limit is reached.
5. This rule is particularly useful when dealing with limits at infinity and infinite limits, allowing for easier evaluation of complex rational functions.

Review Questions

• How does L'Hôpital's Rule help in evaluating limits that result in indeterminate forms?
  • L'Hôpital's Rule provides a systematic way to evaluate limits when direct substitution results in indeterminate forms like 0/0 or ∞/∞. By allowing us to take the derivatives of the numerator and denominator instead, it transforms the problem into one that can often be resolved more easily. If after applying the rule we still encounter an indeterminate form, we can continue using the rule until we reach a determinate limit.
• Discuss a scenario where you would apply L'Hôpital's Rule and explain why it is necessary in that situation.
  • Consider the limit $$ ext{lim}_{x o 0} rac{ an(x)}{x}$$. Direct substitution gives us 0/0, which is an indeterminate form. Applying L'Hôpital's Rule involves taking the derivative of the numerator (sec²(x)) and the derivative of the denominator (1). Evaluating $$ ext{lim}_{x o 0} rac{ ext{sec}^2(x)}{1}$$ gives us a determinate limit of 1. This shows how L'Hôpital's Rule is essential when direct evaluation leads to ambiguity.
• Evaluate the limit $$ ext{lim}_{x o rac{ ext{ extit{π}}}{2}} rac{ an(x)}{ ext{sin}(x)}$$ using L'Hôpital's Rule and analyze its significance.
  • Direct substitution into $$rac{ an(rac{ ext{ extit{π}}}{2})}{ ext{sin}(rac{ ext{ extit{π}}}{2})}$$ results in ∞/1, which is not indeterminate. However, for values approaching $$rac{ ext{ extit{π}}}{2}$$ from either side, we notice both functions grow without bounds due to their respective asymptotes. We could modify our approach to use L'Hôpital's Rule on something like $$rac{ ext{sin}(x)}{rac{1}{ ext{cos}(x)}}$$ if we rewrite it appropriately. The significance lies in demonstrating how L'Hôpital's Rule allows us to navigate through complex trigonometric limits by addressing their undefined behavior at specific points.

Related terms

Indeterminate Form: A mathematical expression that does not have a well-defined value, often seen in limits, such as 0/0 or ∞/∞.

Derivative:

A measure of how a function changes as its input changes; essentially the slope of the tangent line to the graph of the function.

Limit:

A fundamental concept in calculus that describes the value a function approaches as the input approaches a particular point.