Taking the logarithmic derivative of an exponential difference function after applying L'Hospital's RuleCan somebody please...

Jorden Pace

Jorden Pace

Answered

2022-07-04

Taking the logarithmic derivative of an exponential difference function after applying L'Hospital's Rule
Can somebody please explain the following application of L’Hospital’s Rule?
Find the limit:
lim x 0 5 x 3 x x
Solution:
Determining that this function has indeterminate form 0 / 0, we apply L’Hospital’s rule.
Applying L.H. rule, we get lim x 0 5 x ln 5 3 x ln 3 1 = lim x 0 ( ln 5 ln 3 ) = ln 5 3
The part that I am confused on is the application of the natural logarithm that occurs after the first application of L’Hospital’s Rule, namely lim x 0 5 x ln 5 3 x ln 3 1 = lim x 0 ( ln 5 ln 3 ). I am not sure how the derivative becomes what it does, nor do I understand the next step. I do understand lim x 0 ( ln 5 ln 3 ) = ln 5 3 , however.
Thanks!

Answer & Explanation

furniranizq

furniranizq

Expert

2022-07-05Added 20 answers

5 x = e x ln 5
Now use d d x e x = e x and the chain rule:
d d x 5 x = d d x e x ln 5 = ln ( 5 ) e x ln 5 = ln ( 5 ) 5 x
pouzdrotf

pouzdrotf

Expert

2022-07-06Added 4 answers

You're just letting x 0. Since 5 0 = 3 0 = 1, those factors disappear.

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