Calculus 1: L'Hospital's Rule (2 of 25) Example 1

In this video I will use L'Hospital's Rule to solve limit-as-x-approaches-0 of [(3^x-1)/x].

Next video in the series can be seen at:

[someone]

There's an easier way to find (3^x)' assume y = 3^x then we take the natural log for both hand sides so we get In(y) = x In(3) = In(3) x (notice that In(3) is a constant) then we take the derivative for both sides so we get y'/y = In(3) which means y' = y In(3) = 3^x In(3)
#GENERALLY (a^x)' = a^x In(a)

[sthakar]

Neat concept of writing 3 as e^ln3. However, we can use a direct property a^x = a^x ln(a). Using this we can say, 3^x= 3^x (ln3), and as x approaches 0, 3^x(ln3) approaches ln(3) :-). We got the answer in one step.

[Galileosays]

Thanks for this cute limit. By the way L'Hopital is pronounced Lopital, the H is silent. The French family name means "the hospital". So, probably his ancestors were medical doctors at an hospital. In the end, which is a limit, L'Hopital's rule is a nice remedy.

[rajeevmishra]

Its very easy sir thankyou so much for your clear understanding explanation . By the way i am from India

[francis]

I could be wrong but isn't using L'Hopital's rule on a question like this just circular reasoning? Isn't this limit part of the derivative of 3^x?

[PrakashPandey-xjco]

If question is given in under root and is 0/0 form then first we have to rationalize and then apply l hospital rule or directly apply it

[ashishreddy]

It’s better to apply derivative of a^x=a^x log a.its simple and easy...

[joeyborja]

If we used the other rule where d/dx a^x = a ln a (where a is a constant), we would arrive to the same answer.

[simantasaha]

If there tends to infinity then what was there????

[yisraelbenavraham]

I thought the derivative of ln is 1/x. So why isn't the derivative of ln(3)x 1/3?