# Math Help - Real Analysis help needed

1. ## Real Analysis help needed

Hi Im trying to do Q1 and Q2(a) on this past exam sheet and am having alot of trouble . I attached the Pdf cos im not sure how to use LAtex.

Any help is very much appreciated

Q1b Do i need to calculate the partial derivatives using the definition. im confused s the function is defined in 2 parts. If you can calculate the directional derivatives at (0,0) how can you also show that it is NOT differentiable at (0,0)

Q2(a) I get b =Log(m)/Log(e) might be wrong but then i cant see how to use that for the next part.

THANKS!

2. For 2a:

I am going to use ln instead of log.

$\lim_{n\to \infty}n^{\frac{1}{ln(n)}}$

Let $t=ln(n)\rightarrow e^{t}=n$

$\lim_{t\to \infty}\left(e^{t}\right)^{\frac{1}{t}}=\lim_{t\to \infty}e^{1}=e$

If you must use log, then it's base is 10 and the limit will be 10 instead of e.

Then we have $\lim_{t\to \infty}(10^{t})^{\frac{1}{t}}=10$

So, since we have $m=e^{b}$, see what b is now?.

3. so b is ln(m)?

So for part 2a (ii)

Its Lim (4)^1/t and 1/t --> 0 so the limit is 1?

4. ## Quick helo needed with directional derivative question

Hi I still need help with Q 2 b. on the attached PDF. I need to know this for the exam tomorrow.

What i have done so far is calculated the partial derivatives of x and y and i presume i have to do it using some general unit vector and then use the directional deravitive formula and fill it in at the point (0,0)

But then when i fill this in i get 0 as the denominator

OR is it enought to go that far ant not fill in (0,0)