Limit Exercise 2 (Request Hint Only)

[DrToddMatthews]

I need a hint to find the direction from which I should approach this exercise.

  lim  [ sqrt(r)/(r - 9)^4 ]
  r->9

By inspection, I see that -- as r approaches 9 -- the denominator becomes a very tiny positive value, while the numerator becomes closer to 3. This spells infinity.

I'm stuck on the algebraic manipulation(s) to show that the limit is infinity.

:?

 

[Unco]

The denominator tends to zero, the numerator tends to a non-zero number; end of story.

 

[DrToddMatthews]

Ummm ...

Hello Unco:

The denominator tends to zero, the numerator tends to a non-zero number

Isn't that what I posted (in so many words)?

:wink:

end of story.

I'm chuckling as I type this because I'm thinking about what my instructor's response would be if I were to answer an exam question with those three words.

I need to come up with an algebraic simplification using either of the two limit equations for determining the derivative at r = 9.

 

[Unco]

In your first post you asked how to show the limit was infinity, hence my response.

If f(r) was what was inside the limit and you now are looking for f'(9), the curve is discontinuous at x=9 so it will not have a derivative there.

 

[galactus]

\(\displaystyle \lim_{x\to\9}\frac{sqrt{x}}{(x-9)^{4}}\)

=\(\displaystyle \lim_{x\to\9}\sqrt{x}\lim_{x\to\9}\frac{1}{(x-9)^{4}}\)

=\(\displaystyle \sqrt{\lim_{x\to\9}{x}}\lim_{x\to\9}\frac{1}{(x-9)^{4}}\)

=\(\displaystyle \3\lim_{x\to\9}\frac{1}{(x-9)^{4}}\)

Since \(\displaystyle \lim_{x\to\9}\frac{1}{(x-9)^{4}}={\infty}\)

we essentially have \(\displaystyle 3{\infty}={\infty}\)

 

[DrToddMatthews]

I Made A Mistake In My Reply

In your first post you asked how to show the limit was infinity, hence my response.

I was looking for an algebraic method for finding the limit as r --> 9.

... and you now are looking for f'(9), the curve is discontinuous at x=9 so it will not have a derivative there.

You're right. I made a mistake when I said, "the derivative at 9".

Galactus, your response helps. I'll just have to be satisfied with the analytical part to tie it all together.