# Limits problem

1. Sep 25, 2008

### ganondorf29

1. The problem statement, all variables and given/known data

lim (x^2-5x+4) / ((sin(x^1/2) - 2))
x->4

2. Relevant equations

3. The attempt at a solution

Well I factored the top out to be (x-1)(x-4) and if I plug in 4 I get 0 in the numerator. In the denominator, if I plug in 4, I also end up with zero. Im not too sure what Im supposed to do now

2. Sep 25, 2008

### sutupidmath

$$\lim_{x \rightarrow 4}\frac{x^2-5x+4}{sin(\sqrt x-2)}$$

Are u allowed to apply l'hopitals rule? If so, then it will work nicely!

3. Sep 25, 2008

### ganondorf29

No, we cant use l'hopitals rule because we haven't learned it yet.

4. Sep 25, 2008

### sutupidmath

Ok, then here it is what just popped into my head,

i would let $$\sqrt x-2=t$$ so when x-->4, t-->0

now

$$\lim_{x \rightarrow 4}\frac{x^2-5x+4}{sin(\sqrt x-2)}=\lim_{x \rightarrow 4}\frac{(x-1)(\sqrt x-2)(\sqrt x+2)}{sin(\sqrt x-2)}=\lim_{t\rightarrow 0}\frac{t}{sin(t)}(t+4)[(t+2)^2-1]$$

now:

$$\lim_{t\rightarrow 0}\frac{t}{sint}=\lim_{t\rightarrow 0}{\frac{1}{\frac{sint}{t}}=1$$

I think the rest is easy!