# This was on my test

1. Mar 16, 2006

### mohlam12

Hey everyone,
I had a math test yesterday. It was pretty hard... This is one of the limits I wasn't able to do.

$$\lim_{x \rightarrow k} \frac {x\sqrt{x} - k\sqrt{k}}{x^{4}-k^{4}}$$

I tried the Hopital rule, I tried multiplying the whole expression with the denominator. I didn't get to anything better.

Anyone knows how to do this kind of limits ? Thank you!

2. Mar 16, 2006

### arunbg

Hint: Divide numerator and denominator by x-k.
Then use
$$\lim_{x \rightarrow k} \frac {x^n-k^n}{x-k}=nk^{n-1}$$
for both num and den.

3. Mar 16, 2006

### Curious3141

Factorise the denominator. (x^4 - k^4) = (x-k)(x+k)(x^2+k^2). The rightmost two factors can be evaluated immediately at the limit, they become (2k) and (2k^2) respectively, yes ?

Then the limit becomes

$$\frac{1}{(2k)(2k^2)}\lim_{x \rightarrow k} \frac {x^{1.5} - k^{1.5}}{x-k}$$

Now observe that the limit that's left is of the form 0/0, and can be reduced by LH rule. Just differentiate numerator and denominator wrt x. Put x = k into that, simplify the algebra and you're left with an expression in k.

Last edited: Mar 16, 2006
4. Mar 16, 2006

### BerryBoy

Why can't L'Hopital's rule work from the start?

5. Mar 16, 2006

### Curious3141

It can ! Stupid me. Orig poster, disregard my post and just differentiate numerator and denominator to get a single expression in x and set x = k.

6. Mar 16, 2006

### mohlam12

If you use L'Hopital rule in $$\frac {x^{1.5} - k^{1.5}}{x-k}$$, you will get $$\frac {6 \sqrt{k} -6 \sqrt{x}}{4 \sqrt{kx} *(x-k)'}$$
Then what can we do ?

Last edited: Mar 16, 2006
7. Mar 16, 2006

### Curious3141

You're not differentiating correctly, the k is a constant and vanishes from both numerator and denominator.

8. Mar 16, 2006

### mohlam12

!!!!
If only I knew when passing the test!!
Thank you guys

Last edited: Mar 16, 2006
9. Mar 16, 2006

### greytomato

Please note that L'Hopital rule can only be used when your expression is equal to 0/0 or inf/inf. so you need to check your expression each time before you use the rule.

10. Mar 23, 2006

### dextercioby

Multiply both the denominator and the numerator by $x\sqrt{x}+k\sqrt{k}$ and then simplify the fraction by $x-k$.

Daniel.

11. Mar 23, 2006

### Galileo

Or change the variable $u=x^4$ and let $a=k^4$, then $u \to a$ as $x \to k$ and the limit becomes:

$$\lim_{u \to a} \frac{u^{3/8}-a^{3/8}}{u-a}$$
which is the derivative of $f(u)=u^{3/8}$ at u=a.