Question

help

Answer 1

\[\lim_{t \rightarrow 0}\frac{ 4\sin (7t^5) }{ 2t }\]

Answer 2

Use L'Hopital's Rule to evaluate

Answer 3

do you even know what l hopital rule is?...

Answer 4

yes

Answer 5

so you're having trouble with the derivative?

Answer 6

yes

Answer 7

starting with the top,

the derivative of
\[4\sin(7t^5)\]

we need to apply "chain rule"
sound familiar?

Answer 8

yes, but I always mess it up with sin

Answer 9

ok we'll do it step by step

chain rule ,
\[\frac{df}{dt} = \frac{df}{du}\frac{du}{dt}\]

ok so when it comes to sine and cosines, we want "u" to be equal to whatever is within the parenthensis

so u would equal?
u= ?

Answer 10

7t^5

Answer 11

leave it in the form u=7t^5
but yes

ok now going back to chain rule, we need du/dt

so we need to take the derivative of u with respect to t

so u' =?

Answer 12

35t^4

Answer 13

yes, du/dt = 35t^4

write that down somewhere or just remember it, we will need it later

next we need to find df/du
f being the original function

now remember we denoted the stuff within the parenthesis as "u"

so you need to take the derivative of
4 sin(u)
with respect to u

Answer 14

does this make sense to you?

Answer 15

4 cos(u)

Answer 16

and yes it makes sense

Answer 17

k, please tell me when and/or if my instructions become unclear

anyways, yes that's right assuming im not doing it incorrectly in my head

Answer 18

ok anyways
now we have df/du and du/dt

when you multiply them together, what do you get?

Answer 19

I think this is the part I mess up at

Answer 20

dont worry about it,

just multiply the 2 terms we determined above and show me what you get

if you make a mistake, ill point it out

Answer 21

140t^4(cos(u))

Answer 22

then plug in u

Answer 23

yea

Answer 24

and 140t^4(cos(7t^5))

Answer 25

good, now remember that's the top

now we need to determine the derivative of the bottom

Answer 26

2

Answer 27

ok so now the limit looks like \[\lim_{t \rightarrow 0}\frac{140t^4(\cos(7t^5))}{2}\]

Answer 28

can you solve it now?

Answer 29

\[ 70t^4(\cos(7t^5))\]

Answer 30

i didnt mean simplify it, although that still works

take the limit when t approaches 0

Answer 31

0?

Answer 32

yup

Answer 33

I hate calculus problems...

Answer 34

Thank you so much!