L’Hôpital’s rule practice question 7
Evaluate the limit.
Before reaching for l’Hôpital’s rule, substitute into the top and the bottom separately. Is the form really ?
Since , neither the top nor the bottom is 0 at . For a quotient of continuous functions with a nonzero bottom, the limit is found by substituting.
A number, as a fraction or a decimal.
Theory UBC Math 100 §6.1CLP-1 §3.7Key results
Worked solution
Strategy
Check the form first. The top and the bottom are continuous at 1 and the bottom is not 0 there, so the limit is just the value of the quotient.
Working
log 1 = 0. The form is 1/2, which is not indeterminate.
A quotient of continuous functions with a non-zero denominator: substitute.
Using the rule anyway gives a different number, and a wrong one: the rule only holds for 0/0 and ∞/∞.
Answer
The answer is one half.
Check
At x = 0.99 and 1.01 the quotient is about 0.487 and 0.512, either side of 1/2.
Where this goes wrong
Differentiating the top and the bottom out of habit, which gives 3. Substituting first shows the form is 1/2; a form other than 0/0 or ∞/∞ decides the limit directly.
The idea behind it
The L’Hôpital’s rule lesson, a figure to move, in a few steps
Leans on
Evaluating limits by substitution, from Limits
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