# I have a problem in understanding a problem about the limit of a function.

• Aug 10th 2010, 11:46 AM
Real9999
I have a problem in understanding a problem about the limit of a function.
The question is as follows:

Use l'Hopital's rule to evaluate

lim [(a^x - b^x) / (e^ax - e^bx)] when x approaches 0. (Notice: e here stands for exponential symbol).

The answer given by the textbook is (lna - lnb) / (a - b). However, I have problem of getting to this answer. In particular, I don't know how to obtain (a - b) in the denominator.

• Aug 10th 2010, 11:56 AM
yeKciM
Quote:

Originally Posted by Real9999
The question is as follows:

Use l'Hopital's rule to evaluate

lim [(a^x - b^x) / (e^ax - e^bx)] when x approaches 0. (Notice: e here stands for exponential symbol).

The answer given by the textbook is (lna - lnb) / (a - b). However, I have problem of getting to this answer. In particular, I don't know how to obtain (a - b) in the denominator.

to where did u come with your work, show and we'll see where is the problem that u have :D

just to note that :
$\displaystyle \displaystyle \frac {d}{dx} ( e^{ax} )=ae^{ax}$

$\displaystyle \displaystyle \frac {d}{dx} ( a^{x} )=a^{x} \ln {a}$

$\displaystyle e^0 =1$

does this helps ? or ? need to go in that more detailed ?

anyway after using L'Hospital's rule (and knowing derivate's from above ) u can easy come to conclusion that result will be

$\displaystyle \displaystyle \frac {\ln{a}-\ln{b}}{a-b}$
• Aug 10th 2010, 01:28 PM
Krizalid
this can be done by not using that rule, if you want the solution, let me know.
• Aug 10th 2010, 02:18 PM
yeKciM
Quote:

Originally Posted by Krizalid
this can be done by not using that rule, if you want the solution, let me know.

:D but it's clearly stated that he have to use L'Hospital's rule by his concept of question : "Use L'Hospital's rule to evaluate.. " and evaluating this using any another procedure could (although lead to the correct solution) lead to incorrect solution (as for teachers point of view) :D
• Aug 10th 2010, 02:22 PM
Defunkt
Quote:

Originally Posted by yeKciM
:D but it's clearly stated that he have to use L'Hospital's rule by his concept of question : "Use L'Hospital's rule to evaluate.. " and evaluating this using any another procedure could (although lead to the correct solution) lead to incorrect solution (as for teachers point of view) :D

Which is why he did not provide the solution but rather asked him if he wants the other solution :P
• Aug 13th 2010, 01:02 AM
Real9999
Thank you very much for your replies! thanks to all! And sorry for the late reply as I was sick over the past two days...

I still have a problem in understanding how to come to the (a-b) in the denominator. My procedure of doing this question is to differentiate the original function, [(a^x - b^x) / (e^ax - e^bx)], by using the quotient rules. I came to (e^ax - e^bx)^2, and then to (e^ax - e^bx) in the denominator after the entire calculation by the quotient rule. That is, I still do not understand how to go from (e^ax - e^bx) to (a - b).

Sorry for the mess, I will now learn how to type mathematical symbols.
• Aug 13th 2010, 01:03 AM
Real9999
Still have a problem.
Oh, I know how to come to the answer now! The problem I had was that I misunderstood the l'Hopital's rule!!! thanks to all!