L'Hospital Lens
When the top and the bottom of a fraction both approach 0, the limit can be any number. Zoom in to see why the ratio of the derivatives settles it. Then practice the way the exam scores it: two separate limits, the rule applied with limit notation, and the value.
Limit
numerator f(x)
denominator g(x)
dashed: tangent lines at x = a
Zoom in. Close to x = a, each curve looks like its tangent line, so f(x)/g(x) looks like the ratio of the two slopes.
At x = a
Closer and closer
The table, step by step
Find k
Try this
How the exam scores it
- 1. Show the formWrite two separate limits: the numerator → 0 and the denominator → 0. Writing that the limit equals 0/0 does not earn the point. Give the reason: differentiable functions are continuous.
- 2. Apply the ruleTake the limit of (derivative of the top) ÷ (derivative of the bottom). Keep the limit notation. This is not the quotient rule.
- 3. The valueSubstitute, using values from the table or formula.
- Only for 0/0 or ∞/∞If the limit is not indeterminate, substitute directly. The rule would give a wrong answer.
- Still 0/0?Apply the rule again.
Every problem is generated fresh, and every answer is checked by a computer algebra system that finds each limit without using L’Hospital’s Rule. Scores stay in this browser only. Non-calculator. Topic 4.7 Using L’Hospital’s Rule for Determining Limits of Indeterminate Forms.