L’Hôpital’s rule practice question 5
Evaluate the limit.
As , what do the top and the bottom each do? Is the form one the rule applies to?
Both grow without bound: the form is , at infinity, so the rule applies. Differentiate the top and the bottom separately.
The derivatives are and , and .
A fraction, or a decimal.
Theory UBC Math 100 §6.2CLP-1 §3.7Key results
Worked solution
Strategy
The form is ∞/∞ as x → ∞, one of the two the rule settles. One application leaves a quotient whose limit can be read off.
Working
log x grows more slowly than x, so neither the top nor the bottom is held back: an ∞/∞ form.
Differentiate the top and the bottom separately, not the quotient.
The form is no longer indeterminate, so stop.
Answer
The answer is seven halves.
Check
At x = 1000 the quotient is 7006.9/1993.1, about 3.52, and it closes in on 3.5 as x grows.
Where this goes wrong
Letting the logarithms decide because they carry the visible signs. Every power of x outgrows log x, so the coefficients of x set the limit; dividing the top and the bottom by x gives the same 7/2 without the rule.
The idea behind it
The L’Hôpital’s rule lesson, a figure to move, in a few steps
Leans on
Limits at infinity, from Limits
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