Section 4.10
L'Hospital's Rule
When limits fight each other (like or ), nobody wins until we substitute the functions for their derivatives. This powerful shortcut saves us from complex algebra.
1
Local Linearity
If , why can we use derivatives?
Zoom in far enough, and every curve looks like a straight line (its tangent).
So, the ratio of the function heights becomes simply the ratio of their steepness (slopes):
Valid ONLY for 0/0 or forms!
Calculus Microscope
Zoom in to . Notice how the function approaches a height of 1.
Determinate vs. Indeterminate Forms
Indeterminate Forms
Use L'Hôpital's Rule
Determinate Forms
Do Not Use L'Hôpital's Rule
2
Worked Example
Infinity vs. Infinity
Evaluate .
1. Check Form
Top: . Bottom: .
Form is . Apply Rule!
2. Differentiate Top and Bottom
3. New Limit
Conclusion: grows much faster than .
3
Level Up Examples
Example A: Trig
Evaluate .
1. Check Form:
and . Form .2. Apply Rule (Once):
. Still !3. Apply Rule (Again):
.Example B: Indeterminate Powers
Evaluate .
1. Log It:
(Form ).2. Move x to Denom:
(Form ).3. Apply Rule:
Derivative of 1/x cancels out!
.This is the definition of e.
Example C: Subtraction Form
Evaluate .
1. Check Form:
. (Indeterminate, but not a fraction).2. Combine Fractions:
.Now plugging in gives .
3. Apply Rule:
.Example D: Zero Power
Evaluate .
1. Log It:
Let . Then .Form is .
2. Rearrange:
(Form ).3. Apply Rule:
.Since , then .
5
Practice Quiz
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