Jump to content

Menu

math: limits problem (rational expressions) help needed


Recommended Posts

My DS is studying Calc 1.

He is doing finding derivatives by the limits method:

 

f'(x) =  lim             f(x+deltax) - f(x)
          deltax->0    -------------------
                                 deltax

 

problem:

 

                    7
        f(x) =  -------
                 srqt(2x)

 

        after pulling out the constant to save for later
           
               1                        1
             ------           -       ---
      srqt((x+deltax)          srqt(x)
        ----------------------------------
                   deltax
           

       after rationalizing this mess I still have 0/0 indeterminate form  ??

        (tried several approaches)

 

         this is really a rational radical expressions problem (rusty)

 

         can someone take a crack at this -  show steps (posting a picture or via PM since math style above is very painful)

 

   the final answer is easy to figure out the normal derivative way not looking for that

 

   thank you

 

 

   

 

Link to comment
Share on other sites

It works directly, too.

set up the quotient, put on common denominator, square the equation

that leaves an expression that contains one sqrt

move all other terms to the left side to the derivative and square again to get rid of the sqrts.

this gives you a quadratic equation for the derivative which resolves nicely

  • Like 1
Link to comment
Share on other sites

What went wrong when you rationalized? The trick here is to multiply the numerator and denominator by the conjugate of the numerator. (Often in rational expressions, you multiply by the conjugate of the denominator but not here.)

 

You start with (I'm using h instead of deltax for ease of writing):

 

1/h ( 1/sqrt(x+h) - 1/sqrt(x) )

 

Put over one denominator:

 

( sqrt(x) - sqrt(x+h) ) / ( h sqrt(x) sqrt(x+h) )

 

Now multiply top and bottom by sqrt(x) + sqrt(x+h):

 

( x - (x+h) ) / ( h sqrt(x) sqrt(x+h) (sqrt(x) + sqrt(x+h)) )

 

Top simplifies to h, which cancels with the h in the bottom to give:

 

1 / (sqrt(x) sqrt(x+h) (sqrt(x) + sqrt(x+h))

 

And you can easily take the limit of that expression.

  • Like 1
Link to comment
Share on other sites

Join the conversation

You can post now and register later. If you have an account, sign in now to post with your account.

Guest
Reply to this topic...

×   Pasted as rich text.   Paste as plain text instead

  Only 75 emoji are allowed.

×   Your link has been automatically embedded.   Display as a link instead

×   Your previous content has been restored.   Clear editor

×   You cannot paste images directly. Upload or insert images from URL.

 Share

×
×
  • Create New...