Question

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule does not apply, explain why. I shall use the equation editor to post the limit below.

Answer 1

\[\lim_{t \rightarrow 0}\frac{8^t-5^t}{ t }\]

Answer 2

its having the the two numbers with the variable in the exponent that is confusing me

Answer 3

limt/t→0
8^t−5^t/t

Answer 4

do you know the formula for lim x->0 (a^x-1)/x

Answer 5

use it for this question for the elementary method.

Answer 6

but we can solve this using L'Hopital's

Answer 7

that sounds familiar but it could have been a year or two since I used that formula

Answer 8

ok, what about L'Hopital's , know it ?

Answer 9

yes this section is on L'Hopital I take the derivative of the numerator and the derivative of the denominator but for some reason having the two terms to the power of the variable is causing me to mess it up.

Answer 10

derivative of numerator =... ?
derivative of 8^t = 8^t log 8
derivative of 5^t = 5^t log 5
so,derivative of 8^t -5^t= 8^t log 8- 5^t log 5
derivative of denominator =... ?

Answer 11

derivative of numerator =... ?
derivative of 8^t = 8^t log 8
derivative of 5^t = 5^t log 5
so,derivative of 8^t -5^t= 8^t log 8- 5^t log 5
derivative of denominator =... ?

Answer 12

I may have screwed up something very simple here . . . by log do you mean the natural logarithm or a logarithm of base 10

Answer 13

in calculus, if only 'log' is mentioned, its natural log.
outside calculus, 'log' means by default its log to base 10 and 'ln' means natural log
so, here , in calculus its natural log

Answer 14

I don't know why but my textbook and instructor still use ln to represent the natural log but I looked at alot of online stuff for this section and I think I entered the answer as if it was log base ten

Answer 15

Maybe its because the professor is Russian or because I live in Canada and they do things slightly different in one of those regions but it looks like thats what I did thanks guys, sorry for wasting your time

Answer 16

so, your answer will be log (8/5) {if log = ln}
or ln (8/5) / ln 10 (by change of base formula)

Answer 17

welcome :)