Question

Use the limit definition to find the slope of the tangent line to the graph of f at the given point.
f(x) =Squareroot of x + 1; (3, 2). I got 1/6 but it is incorrect for some reason. Can someone please help?

Answer 1

what are you using for the limit definition?

Answer 2

roots are generally easier to work with when expressed as exponents, so I assume the function is f(x) = (x+1)^(1/2)

lim h->0 [f(x+h) - f(x)]/h = lim h=>0 {[(3 + h) + 1]^(1/2) - 2}/h
= lim h=>0 {[(3 + h) + 1]^(1/2) - 2}{[(3 + h) + 1]^(1/2) + 2}/(h{[(3 + h) + 1]^(1/2) + 2})
in other words we multiplied both the numerator and denominator by [(3 + h) + 1]^(1/2) + 2, which is equivalent to multiply by 1 and changes nothing.
= lim h->0 {[(3 + h) + 1]^(1/2) - 2}^2/(h{[(3 + h) + 1]^(1/2) + 2})
= lim h->0 [(3 + h + 1) - 4]/(h{[(3 + h) + 1]^(1/2) + 2})
= lim h->0 h/(h{[(3 + h) + 1]^(1/2) + 2})
Lo and behold, the h's cancel then the limit can be solved by substitution
= lim h->0 1/{[(3 + h) + 1]^(1/2) + 2} = 1/[(4)^(1/2) +2] = 1/4

Answer 3

From the video I watched to follow the steps they kept it as square root until the end and they squared the number.. But I think the limit definition he used was the limit delta x as it approaches 0

Answer 4

still looking for help ?

Answer 5

yes

Answer 6

okay :) may i know where exactly are u stuck

Answer 7

I get 1/6

Answer 8

I'm not sure where I go worng. I followed all the same steps the video did

Answer 9

\[\large f'(x) = \lim \limits_{\Delta x \to 0}~ \dfrac{f(x+\Delta x) - f(x)}{\Delta x}\]

Answer 10

using that definition, right ?

Answer 11

And the way Lars solves it looks nothing like the way th e video does

Answer 12

I see..

Answer 13

Use the limit definition to find the slope of the tangent line to the graph of f at the given point.
f(x) =

x + 1
; (3, 2)

Answer 14

it should show square root of x+1

Answer 15

:( somebody please help...

Answer 16

are you using the limit definition I gave u earlier ?

Answer 17

this one :

\[\large f'(x) = \lim \limits_{\Delta x \to 0}~ \dfrac{f(x+\Delta x) - f(x)}{\Delta x}\]

Answer 18

??

Answer 19

sorry my laptop died, had to plug in. it looks exactly like that except the F'(x)
it's just f(x) and then the rest is the same

Answer 20

okay good, f'(x) is just a notation used for derivative/slope - forget it for now. lets continue :)

Answer 21

I end up the last few steps as